American Mathematical Monthly Geometry ... - FAU math

YIU : Problems in Elementary Geometry 1

American Mathematical MonthlyGeometry Problems 1894 –

Vis171. (Marcus Baker)

In a traingle ABC, the center of the circumscribed circle is O, the center ofthe inscribed circle is I, and the orthocenter is H. Knowing the sides of thetriangle OIH, determine the sides of triangle ABC.

Solution by W.P. Casey: Let N be the nine-point center. From IN =R2 − r and OI2 = R(R− 2r), R and r can be determined. The circumcircleand the incircle, and the nine-point circle, can all be constructed. Then,Casey wrote, “[i]t only remains to find a point C in the circumcircle ofABC so that the tangent CA CA to circle I may be bisected by the circleN in the point S, which is easily done”. [sic] The construction of ABC fromOIH cannot be effected by ruler and compass in general. Casey continuedto derive the cubic equation with roots cosα, cosβ, cos γ, and coefficientsin terms of R, r, and := OH, namely,

x3 − (1 +r

R)x2 +

4(R + r)2 − (2 + 3R2)8R2

x− R2 − 2

8R2= 0.

Given triangle OIH, let N be the midpoint of OH. Construct the circlethrough N tangent to OI at O. Extend IN to intersect this circle again atM . The diameter of the circumcircle is equal to the length of IM . Fromthis, the circumcircle, the nine-point circle, and the incircle can be con-structed. Now, it remains to select a point X on the nine point circle sothat the perpendicular to OX is tangent to the incircle. This is in generalnot constructible.


Here is an example: Let OI = 2√

2α, OH =√

42α, and IH =√

15α,where α =

√17. Then s− a, s− b, s− c are the roots of the equation

x3 − 34x2 + 172x− 2 · 172 = 0.

The roots of this equation are inconstructible.

Vis186. (Marcus Baker)

(Malfatti circles) Inscribe in any plane triangle three circles each tangent totwo sides and the other two circles.

Three solutions were published. The second one, by U. Jesse Knisely,contains a construction:

The third solution, by E. B. Seitz, 1 gives the radii of the Malfatti circlesas

x =(1 + tan β

4 )(1 + tan γ4 )

1 + tan α4

· r2,

and analogous expressions for the other two.

Vis52. (James McLaughlin)

(Inverse Malfatti problem) Three circles, radii p, q, r, are drawn in a triangle,each circle touching the other two and two sides of the triangle. Find thesides of the triangle.

1The very first issue of the Monthly contains a biography of Seitz, written by Finkel.


Solution by E.B.Seitz: write σ = p+ q + r.

a =pσ + (p(

√p+

√q +

√r) −√

pqr)√σ −√

pqr(√q +

√r −√

p)(p√q + q

√p+

√pqσ − r

√σ)(p

√r + r

√p+

√prσ − q

√σ)

(q + r)2.

Vis289. (Christine Ladd)

If R is the radius of the circumscribed circle of a triangle ABC, r the radiusof the inscribed circle, p the radius of the circle inscribed in the orthictriangle, I the center of the inscribed circle of the triangle ABC, and Q thecenter of the circle inscribed in the triangle formed by joining the midpointsof the sides of ABC, show that

QI2 =1

2(7Rp − 6Rr + 3r2 + 2R2).

This certainly is not right! The correct formula should be

QI2 =12(2Rp − 6Rr + 3r2 + 2R2).

Q is the Spieker center X10 and

QI2 =18s

∑(−a3 + 2a2(b+ c) − 9abc).

Also, IGIX10

= 23 . Now,

∑(−a3 + 2a2(b+ c) − 9abc) = −s31 + 5s1s2 − 18s3 = 2s(5r2 − 16Rr + s2).

Therefore, QI2 = 14 (5r2 − 16Rr + s2).


Now, for the orthic triangle, Y Zsinα = 2R cosα. This means that Y Z =

R sin 2α. The semiperimeter of the orthic triangle is therefore

s′ =12R(sin 2α+sin 2β+sin 2γ) = 2R sinα sin β sin γ =

abc

4R2=

4 ·R4R2

=R.

Also,′

= 2cosα cos β cos γ.

The inradius of the orthic triangle is

p =′

s′= 2R cosα cos β cos γ.

Now,

cosα cosβ cos γ =(b2 + c2 − a2)(c2 + a2 − b2)(a2 + b2 − c2)

8a2b2c2

=(−s61 + 6s41s2 − 8s21s

22 − 8s31s3 + 16s1s2s3 − 8s23

8s23

=32r2s2[s2 − (2R + r)2]

8(4Rrs)2

=s2 − (2R+ r)2

8R2.

Therefore, p = s2−(2R+r)2

4R .

s2 = (2R+ r)2 + 4Rp.

Finally,

QI2 =14(5r2 − 16Rr + s2)

=14[5r2 − 16Rr + (2R + r)2 + 4Rp]

=14[6r2 − 12Rr + 4R2 + 4Rp]

=12(2Rp − 6Rr + 3r2 + 2R2).


VIS 274.

If the midpoints of three sides of a triangle are joined with the oppositevertices and R1, . . . , R6 are the radii of the circles circumscribed about the6 triangles so formed, and r1, r2, . . . , r6 the radii of the circles inscribed inthese triangles. prove that R1R3R5 = R2R4R6 and

1r1

+1r3

+1r5

=1r2

+1r4

+1r6.

VIS 282 (W.P.Casey)

Let X, Y , Z be the traces of the incenter. Construct circles through I andwith centers at X, Y , Z. The sum of the reciprocals of the radii of allthe circles touching these circles is equal to four times the reciprocal of theinradius.

G1.941.S941. (B.F.Finkel)

Show that the bisectors of the angles formed by producing the sides of aninscribed quadrilateral intersect each other at right angles.

G2.941.S942;943.

(From Bowser’s Trigonometry) Show that

π

2=(

2 · 4 · 6 · 8 · 10 · · ·1 · 3 · 5 · 7 · 9 · · ·

)2

,

Wallis’ expression for π.


P.H. Philbrick pointed out that this expression was not correct. It shouldbe

π

2=(

2 · 4 · 6 · 8 · 10 · · · (2n)1 · 3 · 5 · 7 · 9 · · · (2n− 1)

)2

· 12n + 1

.

G3.941.

(From Todhunter’s Trigonometry) If A be the area of the circle inscribed ina triangle, A1, A2, A3 the ares of the escribed circles, show that

1√A

=1√A1

+1√A2

+1√A3.

G4.941.S943.

(From Todhunter’s Trigonometry) Three circles whose radii are a, b, and ctouch each other externally; prove that the tangents at the points of contactmeet in a point whose distance from any one of them is√

abc

a+ b+ c.

G5.941.S943. (Adolf Bailoff)

If from a variable point in the base of an isosceles triangle, perpendicularsare drawn to the sides, the sum of the perpendiculars is constant and equalto the perpendicular let fall from either extremity of the base to the oppositeside.


G6.941.S94:159–160. (Earl D. West)

Having given the sides 6, 4, 5, and 3 respectively of a trapezium, inscribein a circle, to find the diameter of the circle. [Note: trapezium meansquadrilateral].

G7.941.S94:160. (William Hoover)

Through each point of the straight line x = my + h is drawn a chord of theparabola y2 = 4ax, which is bisected in the point. Prove that this chordtouches the parabola (y − 2mn)2 = 8a(x− h).

G8.941.S944. (Adolf Bailoff)

If the two exterior angles at the base of a triangle are equal, the triangle isisosceles.

G9.941.S94:160–161. (J.C.Gregg)

Two circles intersect in A and B. Through A two lines CAE and DAF aredrawn, each passing through a center and terminated by the circumferences.Show that CA · AE = DA · AF . (Euclid)

G10.941.S94:161. (Eric Doolittle)

If MN be any plane, and A and B any point without the plane, to find apoint P , in the plane, such that AP + PB shall be a minimum.

G11.941. (Lecta Miller)

A gentleman’s residence is at the center of his circular farm containing a =900 acres. He gives to each of his m = 7 children an equal circular farmas large as can be made within the original farm; and he retains as large acircular farm of which his residence is the center, as can be made after thedistribution. Required the area of the farms made.

G12.S94:198–199. (J.F.W.Scheffer)

Let OA and OB represent two variable conjugate semi - diameters of theellipse

x2

a2+y2

b2= 1.


On the chord AB as a side describe an equilateral triangle ABC. Find thelocus of C.

G13.941. (Henry Heaton)

Through two given points to pass four spherical surfaces tangent to twogiven spheres.

G14.S94:232–233;268–269. (Henry Heaton)

Through a given point to draw four circles tangent to two given circles.

G15.S94:233–234. (Issac L. Beverage)

A man starts from the center of a circular 10 acre field and walks duenorth a certain distance, then turns and walks south - west till he comes tothe circumference, walking altogether 40 rods. How far did he walk beforemaking the turn?

G16.941. (H.C.Whitaker)

Three lights of intensities 2, 4, and 5 are placed respectively at points thecoordinates of which are (0, 3), (4, 5) and (9, 0). Find a point in the planeof the lights equally illuminated by all of them.

G17.942:S94:269. (Robert J. Aley)

Draw a circle bisecting the circumference of three given circles.

G18.942.S94:271. (Henry Heaton)

Through two given points to draw two circles tangnet to a given circle.

G19.942:S94:315. (A. Calderhead)

If any point be taken in the circumference of a circle, and lines be drawnfrom it to the three angles of an inscribed equilateral triangle, prove thatthe middle line so drawn is equal to the sum of the other two.


G20.942. (George Bruce Halsted)

Demonstrate by pure spherical geometry that spherical tangents from anypoint in the produced spherical chord common to two intersecting circles ona sphere are equal.

G21.943. (Charles E. Myers)

A cistern 6 feet in diameter contains 3 inches of water. If a cylinder, fourfeet long and one foot in diameter, be laid in a horizontal position on thebottom, to what height will the water rise?

G22.943.S94:316–317. (J.A.Timmons)

Given the perimeter of a triangle = 100(2s), the radius of the inscribed circle= 9(r), and the radius of circumscribed circle = 20(R); it is required to find(1) the sides of the triangle, (2) the radius of the circle circumscribing thetriangle formed by bisecting the exterior angles of the original triangle, (3)the area of the triangle thus formed, all in terms of R, r, and s.

G23.943.S94:352–353. (E.L.Platt)

The ordinate of the point P of an ellipse is produced to meet the circledescribed on the major axis as diameter at Q. CQ, the straight line joiningQ and the center of the ellipse, is tangent to the circle described on the focalradius of P as diameter. If θ is the excentric angle of P , prove that

sin 2θ =2(2a + b) ± 4

√a(a+ b)

a− b.

G24.943.S94:353–354. (T.W.Palmer)

Two right triangles have the same base, the hypotenuse of the first is equalto 60, of the second 40. The point of intersection of the two hypotenuse isat the distance 15 from the base. Find the length of the base.

G25.943.S94:354–355;434. (L.B.Fraker)

The sides of a quadrilateral board are AB = 7, BC = 15, CD = 21, andDA = 13; radius of inscribed circle is 6.

(1) What are the dimensions of the largest rectangular board that canbe cut out of the given board,


(2) largest square,(3) largest equilateral triangle?

G26.943.S94:355, (J.F.W.Scheffer)

ABCD represents a rectangle, and ABEF a trapezoid which is perpendicu-lar to the rectangle, both figure having the side AB common to eahc other,and ADF and BCE forming two right triangles perpendicular to the rect-angle ABCD. To determine the conoidal surface CDFE so as to satisfy thecondition tha any plane laid through AB will intersect it in a straight line.Also find volume of the solid thus formed.

G27.943.S94:355–356. (Adolf Bailoff)

A line BE, that bisects an angle exterior to the vertical angle of an isoscelestriangle is parallel to the base AC.

G28.944. (Henry Heaton)

Through three given points to pass two spherical surfaces tangent to a givensphere.

G29.944:S94:395. (H.W. Holycross)

If the two angles at the base of a triangle are bisected; and through the pointof meeting of the bisectors a line is drawn parallel to the base, the lengthof the parallel between the sides is equal to the sum of the segments of thesides between the parallel and the base.

G30.944.S94:395–396. (Charles E. Myers)

A circle containing one acre is cut by another whose center is on the cirucm-ference of the given circle, and the area common to both is one - half acre.Find the radius of the cutting circle.

G32.94:162.S95:16–17. (W. Hoover)

If a conic be inscribed in a triangle and its focus moves along a given straightline, the locus of the other focus is a conic circumscribing the triangle.

Proof (Hoover). The product of the perpendicular distances from thefoci of a conic to a tangent is constant. Use trilinear coordinates.


G33.94:317.S95:17. (B.F. Sine)

If a given circle is cut by another circle passing through two fixed points thecommon chord passes through a fixed point.

G40:S95:156. (J.C.Corbin)

r1 + r2 + r3 − r = R4 .

This can be found in Chauvenet’s Geometry.

G42:S95:157–158;189–191.

Steiner-Lehmus Theorem (6 proofs).

G43.S95:80. (J.F.W.Scheffer)

The consecutive sides of a quadrilateral are a, b, c, d. Supposing its diagonalsto be equal, find them and also the area of the quadrilateral.

G45.95:122.S95:274–276. (B.F.Burleson)

Determine the radius of a circle circumscribing three tangent circles of radiia = 15, b = 17 and c = 19.

Descartes’ formula; answer not particularly elegant.

G46.95:158.S95:318–319. (G.E. Brockway)

If an equilateral triangle is inscribed in a circle, the sum of the squares ofthe lines joining any point in the circumference to the three vertices of thetriangle is constant.

G47.95:158.S95:319-320. (J.C. Gregg)

Given two points A and B and a circle whose center is O, show that therectangle contained by OB and the perpendicular from B on the polar of Ais equal to the rectangle contained by OB and the perpendicular from A onthe polar of B.

G48.95:233 (I.J.Schwatt)

The Simson line belonging to one point of intersection of Brocard’s diam-eter of a triangle with the circumcircle of this triangle is either parallel or


perpendicular to the bisector of the angle formed by the side BC of triangelABC and the corresponding side B′C ′ of Brocard’s triangle.

G49.95:233,S96:56–57. (J.C. Williams)

Of all triangles inscribed in a given segment of a circle, with the chord asbase, the isosceles is the maximum.

G50.95:276. (B.F.Finkel)

Draw a line perpendicular to the base of a triangle dividing the triangle inthe ratio of m : n.

G50:95.276. (G.B.M.Zerr)

To construct a trapezoid; given the bases, the perpendicular distance be-tween the bases and the angle formed by the diagonals.

G53.S95.320. (B.F.Finkel)

A pole, a certain length of whose top is painted white, is standing on the sideof a hill. A person at A observes that the white part of the pole subtends anangle equal to α and on walking to B, a distance d, directly down the hilltowards the foot of the pole the white part subtends the same angle. Whatis the length of the white part, if the point B is at a distance b from thefoot of the pole?

G54.95:57. (I.J. Schwatt)2

If through the center of perspective D of a given triangle ABC and itsBrocard triangle A′B′C ′ be drawn straight lines so as to pass through Sa,Sb, Sc (the midpoints of the sides BC, CA, AB) and if SaD1 is madeequal to DSa, SbD2 equal to DSb, and ScD3 equal to DSc then are (1) thefiguresD1O

′AO, D2O′BO,D3O

′CO parallelograms (O andO′ are Brocard’spoints), (2) the triangles D1D2D3 and ABC are equal, and (3) D1A, D2,D3C intersect in S (the midpoint of OO′).

2University of Pennsylvania.


American Mathematical MonthlyElementary Problems, 1932 – 1936

E6.329.S333. (W.R.Ransom)

This construction was given in 1525 by Albrecht Durer, the great engraver,for a regular pentagon ABCDE, and it is still given in books on mechanicaldrawing. The circles are all drawn with the same radius, equal to the givenlength of the side AB, with centers at these points (in order) A, B, Q, C,and E. Calculate the angle ABC to determine whether this is an exact oran approximate construction.

E8.329. (Otto A. Spies)

It is required to construct an inscriptible quadrilateral with ruler and com-pass, given the lengths of the four sides in order.

E11.3210.S334. (W.R.Ransom)

Circumscribed about a circle is an isosceles trapezoid, ABCD, in whichDC < AB, and AD = BC. Two perpendiculars are drawn; DG perpendic-ular to AB at G, and GH perpendicular to AD at H. Show that DA, DG,and DH are the arithmetic, geometric, and harmonic means, respectively,between the pair of parallel sides AB and DC.


E12.3210.S334. (W.F.Cheney)

Two coplanar right triangles, AOC and BOC, have the common hypotenuseOC. Using vector methods, express the vector OC in terms of the vectorsOA and OB.

E15.3210.S337. (Pearl C. Miller)

Prove that if two exrternal angle bisectors of a scalene triangle are equal,then the sines of the three interior half - angles form a geometric progression.By external angle bisector is here meant that segment of the line bisectingthe exterior angle at a vertex of a triangle, intercepted between that vertexand the opposite side of the triangle.

E16.331.S335. (G.A.Yonosik)

Prove that the envelope of the circles whose diameters run from points ona parabola to its focus, is the straight line tangent to the parabola at itsvertex.

E17.331. (Wm Fitch Cheney)

Of all the right triangles whose areas exceed a million square units andwhose three sides are integers without common factor, find that one whoseperimeter is minimum.

E29.333.S33:493–494. (J.Rosenbaum)

The faces of a tetrahedron are congruent triangles whose sides are a, b, andc. If 2S = a2 + b2 + c2, show that the volume is

13

√(S − a2)(S − b2)(S − c2).

E38.335. (J.R.Musselman)

It is well known that the midpoints of the sides of any plane quadrilateralconstitute the vertices of a parallelogram. Determine the most general condi-tion under which the parallelogram becomes (a) a rhombus, (b) a rectangle,and (c) a square.


E39.335. (Maud Willey)

LetCi = 0, (i = 1, 2, 3) be the equaitons of three circles. Prove that thethree circles,

∑3i=1KijCi = 0, (j = 1, 2, 3), have the same radical center as

the three circles Ci = 0.Generalization to 3 and higher dimensions.

E40.33(5)296.S34(1)45. (V.F.Ivanoff)

A variable circular arc of constant length has one end fixed in position anddirection. Find the locus of the other end.

E44.336. (Mannis Charosh)

ABC is an isosceles triangle with AB = AC. ADB is a right triangle withD the vertex of the right angle, on the opposite of AB from C. Angle DABis equal to angle BAC, and DF and CE are perpendicular to AB and ADat F and E respectively. Prove that AF and FB differ by AE.

E45.336. (W.R.Ransom)

The ellipse of minimum area which can be circumscribed about a pair ofequal, tangent circles, passes through the centers of its largest circles ofcurvature, and these centers and the two foci are the vertices of a square.

E47.336. (B.H.Brown)

By methods of elementary plane geometry construct an equilateral trianglehaving a vertex upon each of three general lines in a plane, given the positionof one vertex. Consider the case when the lines are parallel, and also thecase in which the three lines are replaced by three concentric circumferences.What determines the number of solutions in the last case?

E48.337. (Norman Anning)

If the squares of the sines of a set of angles are in harmonic progression, showthat the squares of the tangents of the same angles are also in harmonicprogression.


E52.337. (Moshe Abrahami)

Find the area of a triangle in terms of the altitude, interior angle bisector,and median, all from the same vertex of the triangle.

E56.33(8)491.S34(3)189–190. (Otto Dunkel)

From the base vertices A and B of an isosceles triangle ABC, segments ofstraight lines AL and BM of equal length are drawn to the opposite equalsides. Determine by plane geometry the locus of P , the intersection of ALand BM .

E59.338. (J.H.Butchart)

In the angle ACB of triangle ABC circles are inscribed tangent respectivelyto AC at A and to BC at B. Prove that the chords intercepted on the sideAB are equal.

E62.339. (W.R.Ransom)

Defining a “C−angle” as the figure formed by two internally tangent circles,and its magnitude as the difference of the curvature of those circles, showhow to bisect a C−angle geometrically. (If the circles are tangent externally,the magnitude of the C−angle is the sum of their curvatures). If the circlesare tagnet to the X−axis at the origin O, and cut the circle x2+y2 = 2x alsoat P and Q, show that the magnitude of the C−angle equals the differencebetween the slopes of the chords OP and OQ.

E63.339. (J.Rosenbaum)

The bisectors of the interior angles of the triangle ABC meet the sides inthe points P , Q, R. Prove that the ratio of the area of the triangle PQR tothe area of triangle ABC is

2abc(a+ b)(b+ c)(c + a)

.

E65.33(10).606.S34(5)327–328. (J.M.West)

If the vertices of a triangle taken counterclockwise have the abscissas x1, x2,and x3, and if the slopes of the opposite sides are m1, m2, and m3, then


prove that the area equals

12(x1 − x2)(x1 − x3)(m2 −m3),

as well as either of the two similar expressions obtainable from this by cyclicpermutation of the subscripts.

E66.

Solid geometry

E67.3310. (E.C.Kennedy)

Give a scheme for writing down mechanically the sides of an unlimited num-ber of dissimilar right triangles whose sides are integers. After the first set,the values are to be written down, not merely indicated, without any calcula-tions whatever. No addition, subtraction, multipication, division, involutionor evolution, mental or otherwise, is allowed.

E68.3310. (W.F.Cheney)

In the triangle ABC, D is the midpoint of BC. The equilateral trianglesABP , ACQ and ADR are drawn in the plane of triangle ABC, the verticeof each being listed counterclockwise. Prove that R is the midpoint of PQ.


E70.34(1)44.S34(6)391. (R. MacKay)

Show that the area of a right triangle in terms of the bisector of the rightangle, t, and the median to the hypotenuse, m, is given by the formulas

K =2m2t√

t2 + 8m2 ∓ t

where the upper or lower sign is to be used according as t is the bisector ofthe interior or external angle at the right angle vertex.

E72.34(1)45. (J.M.West)

Given that A+B +C = 180circ, prove that∑cyclic

sinA cos2A sin(B − C) = 0.

E.73.34(1)45.S34(6)393–394. (W.F.Cheney)

Show that there is just one right triangle whose three sides are relativelyprime integers between 2000 and 3000.

Answer: (2100, 2059, 2941).

E75.34(2)103. (C.W.Munshower)

Show that in any plane triangle the product of the sum of the ratios of thesides to the radii of the corresponding escribed circles, and the ratio of thesum of the sides to the sum of the radii of the escribed circles, is equal to 4.

E86.34(3)189. (R. MacKay)

If the faces of a tetrahedron are congruent triangles, prove that the circum-center and the centroid are coincident.

E93.34(5)327.S34(10)630. (H.T.R. Aude)

Find the locus of the centers of the circles in the plane which pass througha given point and are orthogonal to a given circle.

E100.34(6)390. (G.R.Livingston)

In two concentric circles, locate parallel chords in the outer circle which aretangent to the inner circle, by the use of compass only, find the ends of thechords and their points of tangency.


E102.34(6)390. (R. MacKay)

If P is a point on the Euler line of triangle whose sides are a, b, c, one k-thof the distance from the circumcenter O to the orthocenter H, then

OP 2 =9R2 − a2 − b2 − c2

k2,

where R is the circumradius of the triangle.

E107.34(7)447. (J.B.Coleman)

A straight line cuts two concentric circles in the points A, B, C, D inthat order. AE and BF are parallel chords, one in each circle. CG isperpendicular to BF at G, adn DH is perpendicular to AE at H. Provethat GF = HE.

E108.34(7)447. (E.Schuyler)

Show how to construct a triangle when the orthocenter, the incenter andone vertex are given.

E113.34(8)517. (E.T.Krach)

Prove that if three circles are so arranged that their six external tangents arereal (each tangent touching two circles), then the three points of intersectionof the three pairs of corresponding tangents are collinear.

E121.34(8)577. (W.F.Cheney)

The sides of the real triangle ABC are three different positive integers, notwo of which have a common factor. AD is tangent to the circumscribedcircle at A, and meets BC produced at D. PRove that AD, BD, and CDare each always rational, but that one of them can ever be an integer.

E122.34(9)577. (C.A.Rasmussen)

The lines joining the three vertices of a given triangle ABC to a point O inits plane, cut the sides opposite the vertices A, B, C in the points K. L,and M respectively. A line through M parallel to KL cuts BC at V andAK at W . Prove that VM = MW .


E125.34(10)629. (E.Schuyler)

Construct the triangle ABC, given the vertex A and the points of contactof BC produced with each of the escribed circles corresponding to sides ACand AB respectively.

E129.34(10)629. (L. Battig)

In the parallelogram ABCD points E and F are in sides AB and CD re-spectively. AF intersects ED in G. EC intersects FB in H. GH producedintersects AD in L and BC in M . Prove by high school geometry thatDL = BM .

E220.36?.S372. (C.W.Trigg)

If circles be constructed on the sides of a triangle as diameters, show that(a) the common tangent to the circles on two of the sides is the mean

proportional between the segments into which the third side is divided bythe point of contact of the incircle; and

(b) the area of the triangle is equal to the square root of the product ofthe three common tangents and the semiperimeter.

See E296.


E259.3720. (Mannis Charosh)

If the tangents of the angles of a plane triangle form an arithmetic progres-sion, prove that the Euler line is parallel to a side of the triangle.

E260.3720. (C.E.Springer)

Two lines AB and CD of given lengths slide independently along two fixedskew lines. Show that the locus of the center of the sphere through A, B, Cand D is a hyperbolic paraboloid.

E262.3720. (Cezar COsnita)

Find the locus of the center of a circle which so varies that its radical axeswith two fixed circles pass always respectively through two fixed points.


American Mathematical MonthlyElementary Problems, 1940 – 1949

E254.37(1)49. (D.L. MacKay)

Given the vertices B and C, and the altitude from A, construct the triangleABC so that a4 = b4 + c4, where a, b, c are the sides of the triangle.

E257.37(1)49.S37(8)540. (M. Charosh)

Construct the triangle ABC, given the altitude and medain from A, andthe difference b− c of the adjacent sides.

E259.37(2)104.S(). (M. Charosh)

If the tangents of the angles are in arithmetic progression, then the Eulerline is parallel to one side of the triangle.

See also E411, E803.

E262.37(2)104.S37(9)599. (C. Cosnita)

Find the locus of the center of a circle which so varies that its radical axeswith two fixed circles pass through respectively two fixed points.

E263.37(2)104.S37(9)599–600. (D.L.MacKay)

In the triangle ABC, the bisector of angle, the median from vertex B, andthe altitude from vertex C are concurrent. Show that the triangle may beconstructed with ruler and compasses if the lengths of sides b and c aregiven.

E265.37(2)104. (W.F. Cheney)

A right triangle has integer sides without common factor. When each digitis replaced by a code letter, the sides are SSWTVU, PTWTS and RRWWQ.Solve the code and show that the solution is unique.


E268.37(3)175. (J.E. Trevor)

A quadrilateral inscribed in a semicircle consists of three chords and thebounding diameter. Find the radius of the semicircle when the successivechords are of lengths a, b, c. Then particularize when a, b, c are 1, 2, 3 feetrespectively.

E269.37(3)175. (C.W. Trigg)

If a cevian be drawn to a side of a triangle and circles inscribed in the twotriangles thus formed, then

(a) the sum of the cevian and the side to which it is drawn is equal to thesemiperimeter and the segment between the points of contact of the circleswith that side;

(b) the product of the radii is equal to the product of the parts intowhich said segment is divided by the cevain;

(c) if the circles are equal, then the area of the original triangle equalsthe product of the radius by the sum of the cevian and the semiperimeter.

E279.37(5)330.S381. (D.L.MacKay)

Given two sides, construct a parallelogram whose angles equal the anglesbetween its diagonals.

E281.37(5)330.S38:51–52. (W.B.Clarke)

Let the incircle of triangle ABC touch side a, b, c at points D, E, F re-spectively. Call the incenter I. With A as center and AE as radius, swingan arc to cut DI product, inside triangle ABC, at P . Similarly, let arcscentered at B and C cut EI and FI, inside the triangle, at Q and R. LetAP , BQ, and CR meet sides a, b, and c at points J , K, and L respectively.Now prove or disprove the following:

(1) AJ , BK, and CL are concurrent;(2) Triangle AJB and AJC, BKA and BKC, CLA and CLB have their

incircles equal each to each in pairs.3

3(1) The lines are not concurrent. (2) The incircles are equal in pairs.


E285.37(6)384.S378. (D.L.MacKay)

If in triangle ABC, sin2A+ sin2B+ sin2C = 1, prove that the circumcirclecuts the nine-point circle orthogonally.

E293.37(6)479. (J.H.Butchart)

Construct three circles through a point P so that the sum of the directedsegments cut off by the circles on any line through P is zero.

E296.37(7)539. (D.L. MacKay)

D, E, F are the centers of the semicircles constructed on the sides BC, CA,AB as diameters, and exterior to the triangle ABC. IF d and e are thelengths of the common external tangents between the points of contact forthe semicircles D and E, and D and F , construct triangle ABC, given d, e,and angle A.

See E220. (a) d =√

(s− b)(s − c) etc.

E301.37(9). (D.L.MacKay)

If in triangle ABC, B − C = 90circ and S, T are the intersections of theinternal and external bisectors of angle A with the side BC, prove:

(a) sinA = b2−c2

b2+c2,

(b) ST is twice the altitude from A,(c) a2 is the harmonic mean of (b− c)2 and (b+ c)2.

E302.37(9). (F.A.Alfieri)

If A, B, C are the angles of a plane triangle, prove that

cotA+ cotB + cotC =12

(sinA

sinB sinC+

sinBsinC sinA

+sinC

sinA sinB

).

E305.37(10)659. (D.L.MacKay)

If the external angle bisectors at A and B are equal, must the triangle beisosceles?


E307.37(10)659. (V.Thebault)

Locate the point P in the plane of the given triangle such that the trianglePAB, PBC, and PCA may have equal perimeters.

E308.37(10)659. (E.H.Clarke)

Find the triangle which contains an angle most nearly equal to one radian,from among all possible triangles whose sides are integers of one or twodigits.

E311.381. (J.S.Robberson)

The quadrant AOB of a circle varies in size and position, but keeps thesegment QA of one bounding radius fixed. Find the locus of the point P onthe arc AB if it divides that arc in the same ratio as Q divides the radiusAO.

E312.381.S389. (D.L.MacKay)

If the scalene triangle ABC has its external angle bisectors at B and Cequal, show that s−a

a is the geometric mean of s−bb and s−c

c .See E15.3210.S337.(Pearl C. Miller).

E314.381. (Cezar Cosnita)

Find the locus of the center of a variable sphere which cuts each of two (orthree) fixed planes in a circle of constant size.

E361.391.S401. (V.Claudian)

The medians of a triangle ABC cut the nine-point circle of that triangleagain at D, E, and F , respectively. The tangents to this circle at D, E andF meet the corresponding sides of the orthic triangle (with vertices at thefeet of the altitudes of ABC) at the points P , Q, R respectively. Provethat P , Q and R are collinear.

E363.391.S407. (D.L.MacKay)

Construct triangle ABC, given A, a, and ha + c− b.


E367.392.S402. (C.Cosnita)

The point P moves on the circumcircle of triangle ABC, and the bisectorsof angles APC and APB meet AC and AB at Q and R respectively. Showthat QR passes through I, the center of the circle inscribed in the triangleABC. Show also that if PS and PT are perpendicular to PQ and PR andcut AC and AB at S and T respectively, then ST passes through the centerof the escribed circle which touches side BC between B and C.

E370.392.S402. (V.Thebault)

Locate the point P within the irregular tetrahedron ABCD so that each ofthe six planes, each through P and an edge, will bisect the surface of thetetrahedron.


The variable point Q moves on a circle thorugh the fixed point A, and Bis another fixed point in the same plane. The points R and S are the feetof the perpendiculars from A and Q on BQ and AB, respectively. The linethrough B, parallel to RS, meets AQ at P . Find the locus of P .

E374.393.S403. (D.L.MacKay)

What relationship exists between the sides of a triangle ABC if the bisectorof angle A, the median from vertex B, and the altitude from vertex C areconcurrent ? Can the three sides be commensurable if the triangle is notequilateral?

See also E263.37p600, and E3434.914, apparently without published so-lution. See R.K.Guy, My favorite elliptic curve: A tale of two types oftriangles, Amer. Math. Monthly, 102 (1995) 771 – 781.

Solution. (C.W.Trigg) By the solution of E263, b cosA = bcb+c . Hence,

a2 = b2 + c2 − 2bc2

b+ c= b2 − c2 +

2c3

b+ c.

If the triangle has commensurable sides, and if the proper unit of measure-ment is chosen, a, b, c will be integers with no common factor. Moreover, band c must be relatively prime, since any common factor of b and c woulddivide a. Hence c3

b+c cannot be an integer,∗ and the only way to make aan integer is to put b = c = 1, in which case the triangle is equilateral.Therefore in all other cases the three sides are incommensurable.


∗ This is not correct. 2c3

b+c can be an integer not divisible by any commondivisor of b and c. For example, E3434 cites the triangle (13,12,15). Here,cosA = 122+152−132

2·12·15 = 59 = 15

12+15 . Here, 2c3

b+c = 250 is not divisible by 3.

E379.394.S403. (W.E.Buker)

Find a trapezoid whose sides, altitude, diagonals and area are rational.

E380.39p297.S403. (W.F.Cheney)

If the radius of a circle is any odd prime p, there are just two differentprimitive Pythagorean triangles circumscriptible about that circle. Showthat, for each such pair of triangles, (a) their shortest sides differ by one;

(b) their hypotenuses exceed their corresponding longer legs by one andby two respectively;

(c) the sum of their perimeters is six times a perfect square;(d) as p increases without limit, the ratio of their least angles approcaches

2;(e) as p increases without limit, the ratio of their areas approaches 2;(f) the smaller triangle can always be placed inside the larger, so as not

to touch it.

E381.395.S406. (W.B.Clarke)

Show how to construct a square with one corner on each of four generallyplaced straight lines in a plane. How many solutions are there in general ?What constitute special cases ? What happens if the lines are placed askewin space ?

E383.395.S404. (Cosnita)

The diameters from the vertices of the triangle ABC, in the circumscribedcircle, cut the opposite sides in E, F and G respectively. L, M and N arethe respective midpoints of AE, BF and CG. Show that triangle LMN ishomologous to triangle ABC, and that the axis of homology is the orthicaxis of the triangle.


On the lateral surface of any right prism, find the length of the shortestroute from end to end on one lateral edge, winding n times round the prism


on the way.

E391.397.S406. (J.Travers)

If P is a point inside a square ABCD, so situated that PA : PB : PC = 1 :2 : 3, calculate the angle APB. Use only the methods of Euclid, Book I.

See also E3208, MG1147, MG1418.932.S942.

***

Given three positive numbers a, b, and c, to construct if possible, a squareABCD, together with an interior point P such that

AP : BP : CP = a : b : c.

Solution. Let Q be a point outside the square such that CBQ = ABPand BCQ = BAP . Then ABP ≡ CBQ. It follows that and BPQis a right isosceles triangle, with PBQ = 90, and PQ =

√2b. Clearly,

then, a,√

2b, and c should satisfy the triangle inequality for such a squareto exist.

If this condition is satisfied, we start with a triangle CPQ with CP = c,CQ = a, and PQ =

√2b. Outside the triangle, erect a right isosceles PBQ

with a right angle at B. The square BCDA on the side of BC containingP satisfies the requirement AP : BP : CP = a : b : c.

Let x be the length of BC. By the cosine formula,

cosPBC =x2 + b2 − c2

2bx, cosQBC =

x2 + b2 − a2

2bx.

Since these two angles are complementary,

(x2 + b2 − c2)2 + (x2 + b2 − a2)2 = (2bx)2;. . . . . . . . .[2x2 − (a2 + c2)]2 = 4(a2 + c2 − b2)b2 − a4 − c4 + 2c2a2. (∗)

If there is nothing wrong here, the right hand side must be nonnegative.Clearly, this quartic form cannot be positive definite. It must have some-thing to do with the condition that a,

√2b, and c form a triangle. Indeed,

it is 162, being the area of triangle CPQ. (One easy way to see this isto replace by

√2b by b′ so that the quartic form becomes

2a2b′2 + 2b′2c2 + 2c2a2 − a4 − b′4 − c4.


It is easy to that this is zero upon the substitution b′ = c + a. It followsthat c+ a− b′ is a factor of this symmetric polynomial; so are a+ b′ − c and−a+ b′ + c. The remaining linear factor must be a+ b′ + c, and indeed theabove polynomial is

(a+ b′ + c)(−a+ b′ + c)(a− b′ + c)(a + b′ − c).

Now it is easy to recognize this as 162).Note that 2x2 = AC2 > a2 +c2 since P is an interior point of the square.

It follows that the length of a side of the square is

x =√

12(a2 + c2 + 4).

The distance DP := d can also be determined easily. If, in the aboveconsideration, we replace every occurrence of b by d, we should arrive at thesame square. This means that in (*) above,

(a2 + c2 − b2)b2 = (a2 + c2 − d2)d2;. . . . . . . . .(d2 − b2)(d2 − a2 − c2 + b2) = 0.

From this, it follows that

d =√a2 + c2 − b2.

How about the case when P is outside the square ?

E394.397.S406. (N.A.Court)

If the lines AM , BM , CM joining any point M to the vertices A, B, C ofa tetrahedron ABCD meet the repectivel opposite faces in the points P , Q,R, and if the lines DM , DP , DR meet the face ABC in the points S, X,Y , Z, prove that (both in magnitude and in sign)

DM

MS=

12

(DP

PX+DQ

QY+DR

RZ

).

E395.397.S406. (Starke)

In high school geometry texts and elsewhere one frequently meets the state-ment that the reason for the straightness of the crease in a folded piece ofpaper is that the intersection of two planes is a straight line. This is fal-lacious. What is the correct reason ? Solution. (L.R.Chase) Let P , P ′ be


two points of the paper that are brought into coincidence by the process offolding. Then any point A of the crease is equidistant from P , P ′, since thelines AP , AP ′ are pressed into coincidence. Hence, the crease, being thelocus of such points A, is the perpendicular bisector of PP ′.

E396.398.S407. (D.L.MacKay)

Given a triangle ABC, construct a point X such that the three lines drawnthrough X, each parallel to a side of the triangle and limited by the othertwo sides, are equal.


Given a triangle ABC, let O be the circumcenter, A′ the projection of A onBC, M any other point of BC, and B1, C1 the respective projections of B,C on AM . Let lines through M , parallel to A′C1 and A′B1, meet AC andAB in points P and Q respectively. Prove that the lines PQ and OM areperpendicular.

E400.398.S407. (H.S.M.Coxeter)

Show how to dissect a regular hexagon by straight cuts into the smallestpossible number of pieces which can be reassembled to form an equilateraltriangle (of the same area).

E405.401.S409 (J.Travers)

Construct points P and Q on the respective sides AB and BC of a giventriangle ABC, so that AP = PQ = QC.

E407.401(correction 406).S409. (V.Claudian)

Let A′, B′ C ′ be the feet of the altitudes of a triangle ABC, and H theorthocenter. Let the parallels through H to B′C ′, C ′A′, A′B′ meet BC,CA, AB in DD, E, F respectively; and let the parrllels through H to BC,CA, AB meet B′C ′, C ′A′, A′B′ in D′, E′, F ′. Prove that the six points D,E, F and D′, E′ F ′ lie respectively on two parallel lines, perpendicular tothe Euler line.


E409.401.C406. (V.Thebault; withdrawn)

Consider a hexagon whose vertices are the ends of three diameters of a circle.Show that the sum of the products of the distances of a variable point onthe circle from pairs of opposite sides of the hexagon is constant. (False).

E410.402.S.()

What are the smallest positive integers a, b, c which are the sides of a trianglewhose medians are also integers?

(Partial solution by W.E.Bueker) The problem of finding triangles whosesides and medians are integers is an old one. (See, Dickson’s History, vol.2,pp.202–205). Particular solutions were obtained by Euler and rediscoveredmany times, the simplest being 174,170,136 for the sides and 127, 131, 158for the medians. (A recent account is to be found in Alliston, MathematicalSnack Bar, pp.24–25.) While these investigations do not seem to prove thatthe above solution is the smallest one, I suggest that otherwise the problemis scarely elementary.Editor’s note: The squares of the medians are easily seen to be

−a′2 + 2b′2 + 2c′2, −b′2 + 2c′2 + 2a′2, −c′2 + 2a′2 + 2b′2,

where a′ = a2 , b′ = b

2 , c′ = c2 . Euler (Operr postuma, vol.1), 1862, pp.102–

103) observed that the values

a′ = (m+ n)p− (m− n)q,b′ = (m− n)p+ (m+ n)q,c′ = 2|mp− nq|

make the third median 2(np+mq), and that the other medians are integerstoo if p = (m2 +n2)(9m2−n2), and q = 2mn(9m2 +n2), m, n being integerssubject to certain inequalities. Discarding any common factor of p and q, weobtain the following simple cases. (But other expressions for p and q mightprovide hitherto unknown solutions).

m n p q Halfsides Medians1 2 25 52 127, 131, 158 261, 255, 2042 1 175 148 377, 619, 404 975, 477, 9423 1 200 123 277, 446, 477 881, 640, 5692 3 13 20 85, 87, 68 131, 127, 1585 3 68 65 207, 328, 145 463, 142, 529


E411.403.S4010. (J.H.Butchart)

Prove that, if the sides of a triangle form an arithmetic progression, the linejoining the centroid to the incenter is parallel to one side.

See also E803 and E259.

E415.403.S4010. (C.Cosnita)

In a triangle of sides a, b, c prove that the distance from the centroid to theincenter I is given by the formula

3(a+ b+ c)GI =√∑

a2(b− c)2 −∑

(b2 + c2 − a2)(c− a)(a− b).

E417.404.S411.(J.F.Kenney)

Let E be any point outside a circle, ABE the diameter through E, andCDE any chord through E. In the triangle BCE, show that the side CE isdivided by D into segments CD and DE whose ratio is less than the ratioof the angles E and C.

E418.404.S411.(W.E.Buker)

Find triangles with rational sides and angle bisectors. Solution. (E.P.Starke)This problem has already been solved as part of E331. Following the nota-tion and analysis given there, the internal angle bisectors have lengths

2bc

b+ ccos

A

2, 2

ca

c+ acos

B

2, 2

ab

a+ bcos

C

2.

The sides being rational, it is necessary and sufficient that the three half-angles have rational cosines. It is then easy to establish that they musthave rational sines. [Denote the incenter by I. We may assume IA, IB, ICrational. Now,

AIB = π − (A+B

2).

The area of triangle AIB being

12IA · IB · sinAIB =

12IA · IB · cos C

2,

is rational. Similarly, the areas of triangles IBC and ICA are rational. Itfollows that the sines of A

2 , B2 , C

2 are all rational.]


It is then easy to establish that they must have rational sines. [Denotethe incenter by I. We may assume IA, IB, IC rational. Now,

AIB = π − ((A+B

2).

The area of triangle AIB being

(12IA · IB · sinAIB = (

12IA · IB · cos(C

2,

is rational. Similarly, the areas of triangles IBC and ICA are rational. Itfollows that the sines of A

2 , B2 , C

2 are all rational.]

Theorem

The internal angle bisectors of a triangle with rational sides are all rationalif and only if tan A

4 , tan B4 , and tan C

4 are all rational.

Theorem

If the sides and internal angle bisectors are rational, so also are the externalangle bisectors, the altitudes, the area, and the five radii.

Let u1 := tan A4 etc. It is easy to see that

1 − u1u2 − u2u3 − u3u1 = u1 + u2 + u3 − u1u2u3,

from whichu3 =

1 − u1 − u2 − u1u2

1 + u1 + u2 − u1u2.

Two positive rational numbers u1, u2 < 1 determine u3 < 1 if and only ifu = (1 + u1)(1 + u2) < 2. In this case,

0 < u3 =2 − u

u− 2u1u2< 1.

These determine

t1 =2u1

1 − u21

, t2 =2u2

1 − u22

, t3 =2u3

1 − u23

,

and give Heronian triangles with rational angle bisectors.

Triangles with rational angle bisectors


(u1, u2, u3) (t1, t2, t3) (a, b, c) (wa, wb, wc)(12 ,

14 ,

113 ) 4

3 ,815 ,

1384 ) (289, 250, 91) (27300

341 , 464138 , 20400

77 )(13 ,

14 ,

29) (3

4 ,815 ,

3677) (289, 250, 231) (400

481 ,11781

52 , 17007 )

(12 ,

15 ,

18) (4

3 ,512 ,

1663) (169, 125, 84) (12600

209 , 26208253 , 975

7 )(13 ,

15 ,

311 ) (3

4 ,512 ,

3356) (169, 125, 154) (30800

279 , 48048323 , 2600

21 )(14 ,

15 ,

514 ) ( 8

15 ,512 ,

140171 ) (338, 289, 399) (101745

344 , 248976737 , 2652

11 )

See also E331.39p172..

E420.404.S411.(V.Claudian)

Let M be the point of intersection of the diagonals of a quadrangle inscribedin a circle with center O. Let parallels through M to the four sides meetthe respective opposite sides at P , Q, R, S. Prove that these four points arecollinear, that their line is perpendicular to OM , and that analogous resultshold for a cyclic hexagon whose three main diagonals concur at a point M .

E421.405.S412.(N.A.Court)

Given four spheres having a point in common, construct a secant throughthis common point, meeting the spheres again in points P , Q, R, S so thatwe shall have, both in magnitude and in sign,

PQ : PR : PS = u : v : w,

wehre u, v, w are given.

E423.405.S412.(C.W.Trigg)

Squares are constructed on the sides of a right triangle ABC. Denote thecentroid of the square on BC and exterior to ABC by A′, and the centroidof the square on BC and “interior” to ABC by A′′. Use correspondingnotation for centroids of the other four squares. Show that

(i) the centroids of ABC, A′B′C ′, A′′B′′C ′′ coincide;(ii) A′B′C ′ and A′′B′′C ′′ are never equilateral;(iii) two vertices of A′B′C ′ (or A′′B′′C ′′) fall on an altitude of A′′B′′C ′′

(or A′B′C ′) and the third vertex falls on the side to which that altitude isdrawn;

(iv) one side of A′B′C ′ (or A′′B′′C ′′) and the altitude to that side areequal; the foot of this altitude divides the side into segments proportionalto the legs of the right triangle;


(v) the sum of the areas of A′B′C ′ and A′′B′′C ′′ equals one - half thearea of the square on the hypotenuse;

(vi) the difference of the areas of A′B′C ′ and A′′B′′C ′′ equals twice thearea of triangle ABC.

E426.406.S413.(V.Thebault)

Find the locus of a point whose polar planes with respect to four givenspheres are concurrent, and the locus of the point of concurrence.

E432.407.S414. (C.W.Trigg)

If a and b are the radii of two spheres, tangent to each other and to a plane,show that the radius of the largest sphere which can pass between them isgiven by the formula √

x =√a+

√b.

E434.407.S415.(D.Arany)

Let F1 and F2 be the foci of a variable ellipse, of major axis 2p, inscribed ina given triangle whose orthocenter is H. Prove that 4p2 −HF)12 −HF 2

2 isconstant.


For what kind of tetrahedron does the Monge point lie on the circumsphere? (The Monge point lies on planes perpendicular to the edges through themidpoints of the respective opposite edges).

E439.408.S416. (J.H.M.Wedderburn)

ABC is a triangle; lines are drawn external to it, parallel to AC and BC atdistances which bear a fixed ratio to the lengths of AC and BC, respectively,making a parallelogram of which CD is one diagonal. If the length of CDis kept constant, show that the locus of C is obtained as follows. Draw twoequal circles with centers A and B, and let a line, equal in length to thediameter of the circles, slide with its ends on the two circles; then C is on thelocus of the midpoint of this line. (The radius of the circles is determinedby any one point on the locus).


E441.409.S415. (D.L.MacKay)

Given ED, construct an isosceles triangle ABC, with apex C, so that E lieson the altitude CD, and two perpendicular transversals drawn through Edivide the area of the triangle into four equal parts.

E443.409.S417. (N.A.Court)

(a) Two triangles, one inscribed in the other, are in perspective. Provethat on a parallel to the axis, the center of perspective trisects the interceptbetween any pair of corresponding sides.

(b) Two tetrahedra, one inscribed in the other, are in perspective. Provethat on a line parallel to the plane of perspective, the center of perspectivequadrisects the intercept between any pair of corresponding faces.

E447.4010.S418. (V.Thebault)

Find the locus of the center of a variable sphere which passes through agiven point and touches two given planes.

E453.411.S418.(N.A.Court)

Given three skew lines a, b, c, for what positions of a point M will theharmonic inverses of M with respect to the pairs b and c, c and a, a and bbe coplanar with M ?

E455.411.S418.(V.W.Graham)

Given a fixed straight line l and a fixed point P outside it, consider twovariable points Q and R on l, such that QPR is constant. Let S be thepoint in which l meets the bisector of this angle, and let C be the center ofthe circle PQR. ORive that CS passes through a fixed point.

E457.41(2)148.S41(9)636–637.(V.Thebault)

Construct three circles which have a common point and which are such thateach touches two sides of a given triangle, the six points of contact beingconcyclic.


Since the circle cuts equal chords on the sides, its center is the incenter.It follows that the pedal of I on the side is the midpoint of the chord, andis on the radical axis of two of the circles. The radical axis is therefore theGergonne cevian. From this the common point is the Gergonne point.

Note: the points of the tangency are the isoscelizers of the Gergonnepoint.

See also E527.426.

E459.412.(V.Claudian)

Show that the altitude and ex-radii of any triangle satisfy the followingrelations:

∑ h2a(rb + rc)

rbrc(ha + 2ra)= 2;

∑ rbrc(rb + rc)(ha + 2ra)

=12.

E463.413.S421. (N.A.Court)

Determine the locus of the trilinear pole of a given line with respect to thetriangle along which a variable plane through the line cuts a given trihedralangle.


In a given triangle, show that the radical axes of the circumcircle with therespective circles whose diameters are the three medians meet the corre-sponding sides in three collinear points.

The line containing the three points is the trilinear polar of the ortho-center. Clearly, this is also the radical axis of the circumcircle and thenine-point circle. The orthic axis is therefore perpendicular to the Eulerline.

See also E507, E568.

E469.414.S422.(V.Claudian)

∑ a2(b2−c2)ra(r2

b−r2

c)= 4R.


E470.414.S422. (W.E.Buker)

Circle I has its center on another circle J . They intersect at A and C.From any point B on J , draw BC intersecting I again at D. Prove thatBD = BA.

E473.414.S423. (N.A.Court)

Two variable transversal planes PQR, P ′Q′R′, reciprocal with respect to agiven tetrahedron DABC meet the edges DA,DB,DC in the pairs of pointsP , P ′; Q, Q′; R, R′. Show that the line of centers of the two spheres DPQRand D′P ′Q′R′ passes through a fixed point. (Two transversal planes aresaid to be reciprocal with respect to a tetrahedron if their traces on eachedge are equidistant from the midpoint of the edge. See Court’s ModernSolid Geometry, p.122, Art. 354.

E475.415.S423. (J.Goodfellow)

Let the diameter AB of a circle S meet a perpendicular chord HH ′ at O.Take points C and D on AB, such that CO = OB and OD = OH. Let Gbe one of the points of intersection of S with the circle on CD as diameter.Show that we have approximately

OG3 = AO ·OB2.

How close an approximation does this construciton provide for the classicalproblem of duplicating the cube ?

E476.416.S423. (A.H.Stone)

Show that it is possible to fit together six isosceles right triangles, all ofdifferent sizes, so as to make a single isosceles right triangle.

E477.416.S423. (Thebault)

Consider four spheres (Si), i = 1, 2, 3, 4, whose centers are the vertices ofa tetrahedron S1S − 2S − 3S − 4. Let (G1) be the sphere whose centeris the centroid of the face S2S3S4 and which passes through the points ofintersection of spheres (S2), (S3), (S4). Defining (G2) and (G3) similarly,prove that the three spheres (G1), (G2), (G3) intersect on the radical axisof (S1), (S2), (S3).

A similar problem for three circles was discussed in Mathesis, 1891, p.238.


E479.416.S423. (D.Arany)

In the plane of a given triangle ABC, find the locus of a point from whichthe sides BC and CA subtend equal angles.

Notes by N.A.Court: This was first formulated by Steiner for two seg-ments having no common end, and for two equal or suppementary angles.Steiner states without proof that the locus consists of two circular cubics,(Crelles J. 45 (1853) p.375). It was further discussed by P.H.Schoute in thesame journal, vol. 99, 1886, p.98. G. de Longchamps solved the problemin the Journal de Mathematiques Speciales, ser. 2, vol.5, 1886, p.39., andconsidered the special case when the two segments are colliner in Journal deMath. Elementaires, ser.2, vol.5, 1886, p.16.

The question was proposed in Nouvelles Annales de Math, ser.3, vol.3,1884, p.351, and was solved some thirty years later by H.Brocard in the samejournal, ser.4, vol.15, 1915, p.18. The problem was solved both analyticallyand synthetically in Amer. Math. Monthly, , 22 (1915), pp.20–22.

E480.416.S423. (D.E.Lynch)

Construct a pentagon whose sides and diagonals are all commensurable. (Fordefiniteness, suppose there are four equal sides, and three equal diagonals).

E481.41p480.S424. (J.A.Todd)

Let ⎛⎝x1 y1 z1x2 y2 z2x3 y3 z3

⎞⎠ and

⎛⎝X1 Y1 Z1

X2 Y2 Z2

X3 Y3 Z3

⎞⎠

be two matrices of nonvanishing numbers, the elements of the second beingthe cofactors of the corresponding elements of the first. Prove that therelation

det

⎛⎝x

−11 y−1

1 z−11

x−12 y−1

2 z−12

x−13 y−1

3 z−13

⎞⎠ = 0 implies det

⎛⎝X

−11 Y −1

1 Z−11

X−12 Y −1

2 Z−12

X−13 Y −1

3 Z−13

⎞⎠ = 0.

Interesting solution via trilinear polarity.

E483.417.S424. (N.A.Court)

Show tha the four spheres having two points in common and each pass-ing through a vertex and the foot of the corresponding altitude of a givenorthocentric tetrahedron form a coaxal pencil.


E485.417.S424. (J.Goodfellow)

Let AOC be an obtuse angle, A and B on a circle with center O. Take Fand G on the minor arc AB, in directions perpendicular to OB and OA.Take D on the major arc AB so that AOD is an equilateral triangle. TakeH on AD, and J on BD, so that HJ is equal and parallel to FG. JoinFH, and produce to meet the circle again at K.Show that the arc AK isapproximately one-third of the arc AB.

E486.418. (J.M.Andreas)

Quadrilateral ABCD has a right angle at A. The angles at B and C arebisected by the diagonals BD and CA. Is the quadrilateral necessarily asquare ?


Prove that if the orthocenter of a triangle is conjugate to the three verticeswith regard to the incircle and two of the excircles, respectively, then thesethree circles touch the respective sides of the orthic triangle, and conversely.

E489.418.S425. (H.Eves)

Let A0, Am, Ah be the areas of the lower base, midsection, and the upperbase of a prismatoid. If Ah = A0, prove that

(1) sections equidistance from the midsection are equal in area;(2) the midsection bisects the volume of the prismatoid;(3) if Am = A0, all sections have the same area;(4) if Am = A0, Am is the maximum or minimum section.

E492.41p635.S426. (N.A.Court)

Given a tetrahedron ABCD and a point M , prove that th etangent planesat M , to the four spheres MBCD, MCDA, MDAB, MABC, meet therespective faces BCD, CDA, DAB, ABC in four coplanar lines.

E493.419. (N.A.Court)

Given a tetrahedron ABCD and a point M , prove that the tangent planes,at M , to the four spheres MBCD, MCDA, MDAB, MABC meet therespective faces BCD, CDA, DAB, ABC in four coplanar lines.


E495.419.S426. (D.Arany)

If x, y, z are the barycentric coordinates of a point Q with respect to atriangle ABC, show that, for any point P in the same plane,

xAP 2 + yBP 2 + zCP 2 = xAQ2 + yBQ2 + zCQ2 + (x+ y + z)PQ2.

E497.4110.S427. (V.Thebault)

The sides of a triangle A′B′C ′ of constant size, remain parallel to thoseof a fixed triangle ABC, and form with it three more triangles and threepentagons. Show that the position of A′B′C ′ which minimizes the sum ofthe areas of these triangles makes the areas of the three pentagons all equal.

E499.4110.S427. (D.H.Browne)

Two intersecting circles (A) and (B) have centers mutually external. Twoother circles (C) and (D), orthogonal to (Q) and (B) respectively, are drawnthrough the points of intersection. Show that the two common tangents of(C) and (D) are concurrent with the two common tangents of (A) and (B).

E501.421.S428. (D.Arany)

If A, B, C, I, J , X are six points on a conic, while L, M , N are pointson the respective sides BC, CA, AB of the triangle ABC, and if furtherthe three pencils L(BXIJ), M(CXIJ), N(AXIJ) are projectively related,prove that the points L, M , N are collinear.

E503.421.S428. (N.A.Court)

Through a point M lines are drawn meeting the pairs of opposite edges ofa given tetrahedron in the pairs of points U , X; V , Y ; W , Z. Prove that ifM bisects each of the three segments UX, V Y , WZ, it coincides with thecentroid of the tetrahedron.

E507.422. (V.Thebault)

In an orthocentric tetrahedron with orthocenter H and circumcenter O,show that the radical planes of the circumsphere with the respective sphereswhose diameters are the four medians, meet the Euler lines of the corre-sponding faces in four points lying in a plane perpendicular to OH. See alsoE467.


E513.42p195.S431. (N.A.Court)

A line revolves about a fixed point in such a manner that the segmentintercepted on it by two intersecting planes has its mid-point in a thirdgiven plane. Show that the locus of the variable line is a cone of the seconddegree.

E515.423.S431. (H.T.R.Aude)

Find all triangles with integral sides which have one side equal to 16 unitsand the cosine of an adjacent angle equal to −1

4 .


If two tetrahedra have equal areas for corresponding faces, do they neces-sarily have the same volume?

E521.425.S431. (J.R.Musselman)

(a) On the sides BC, CA and AB of a triangle ABC, construct externallyany two directly similar triangles CBA1 and ACB1. Show that the mid-points of three segments BC, A1B1, CA form a triangle directly similar tothe given triangles.

(b) On BC externally, and on CA internally, construct any two directlysimilar triangles CBA1 and CAB1. Show that the midpoints of AB andA1B1 form with C a triangle directly similar to the given triangles.

Solution by Howard Eves: The two theorems are very special cases ofthe fundamental theorem concerning two directly similar figures: If the linesjoining corresponding points of two directly similar figures be divided pro-portionally, the locus of the point of division will be a figure directly similarto the given figure.

E523.422.S432. (N.A.Court)

With the vertices of a given orthocentric tetrahedron (T ) as centers, spheresare drawm orthogonal to a given sphere (M) concentric with the polar sphereof (T ). Show that the radical planes of (M) with the four spheres consideredform a tetrahedron which is orthocentric, and that its orthocenter coincideswith that of (T ).


E525.425.S432. (M.Kraitchik)

Find parallelpipeds with commensurable edges and diagonals.

E526.426.S432. (R.C.Yates)

Find the locus of P if the angles formed by the tangents from P to two fixedcircles are equal.


Show that the sum of the radii of the circle C1, C2,C3 of E457.412S419 isequal to the diameter of the incircle, and that the sum of the radii of thethree analogous circles whose centers are exterior to the segments AiI isthree times as great.

E529.42p404.S433.(J.Rosenbaum)

Construct an irregular hexagon which shall be both inscriptible and circum-scriptible.

E530.426.S433. (P.D.Thomas)

Is there a sphere orthogonal to the six radical spheres determined by fourgiven spheres whose centers are not coplanar?

E533.427.S434. (N.A.Court)

Prove that, if an orthocentric group of points occurs as a section of anorthocentric group of lines, then the plane of section is perpendicualr to oneof the lines.

E535.427.S434.(A.H.Stone)

Let A′, B′, C ′ be three points on the circumcircle of a triangle ABC, whoseSimson lines with respect to ABC all meet in a point O. Prove that theSimson lines of A, B, C with respect to the triangle A′B′C ′ concur at thesame point O.

Eves gave a more general lemma and refer to Art. 338 of Johnson.A necessary and sufficient condition for the Simson lines α, β, γ to concur

is that α ⊥ BC, β ⊥ CA, and γ ⊥ AB.



Let L, M , N and L′, M ′, N ′ be the orthogonal projections of a point Pon the sides and the corresponding altitudes of a given triangle. Show thatthe lines LL′, MM ′, NN ′ are in general concurrent, and find the locus ofP when they are parallel.

Answer: The locus is the nine-point circle.Eves generalized this an affine problem.

E539.428.S436,487. (H.Eves)

Give a ruler construction for finding the centers of three given linearly in-dependent circles, no two of which are intersecting, tangent, or concentric.

E540.428.S436. (L.M.Kelly)

Can the radius of the sixteen-point sphere ever be one-half of the circumra-dius of the tetrahedron?

E541.429.S436. (J.Rosenbaum)

Given a regular polygon of n sides, n > 4, design a quadrilateral Q, suchthat (i) it shall be possibel to fit 2n of the Q’s to the polygon to form aregular polygon of n sides, (ii) it shall be possible to fit 2n additonal Q’s tothe new polygon to form a still larger third regular polygon of n sides.

E543.429.S436. (N.A.Court)

Find a point whose polar planes for three given spheres (with non-collinearcenters) are mutually perpendicular. Show that the problem may have twosolutions. When will they be real?

E544.429.S436. (E.P.Starke)

Show that it is possible to construct a tetrahedron such that the length ofevery edge, the area of every face, and the volume all are integers.

E545.429.S436. (A.H.Stone)

Starting with a point P on the side BC of a triangle ABC, mark Q on ABwith BP = BQ, R on CA with AR = AQ, P ′ on BC with CP ′ = CR, Q′


on AB with BQ′ = BP ′, and so on. Prove that the construction closes, i.e.,that CP = CR′, and that the six points P , Q, R, P ′, Q′, R′ are concyclic.

The center of the circle is always the incenter of the triangle.

E547.42p683.S439. (V.Thebault)

A diameter d of the circumcircle of an equilateral triangle ABC cuts thesides BC, CA, AB in points D, E, F . Prove that the Euler lines of thethree triangles AEF , BFD, CDE form a triangle symmetrically equal toABC, the center of symmetry lying on d.

E553.431.S439. (N.A.Court)

If two of the four circles of intersection of two spheres with two planes arecosphereical, prove that the remaining two circles are likewise cospherical.


A sphere (S) of constant radius rolls on a fixed sphere (O) in such a wayas to pass through a fixed point A. Determine the loci of the centers ofsimilitude of the spheres (S) and (O).

E558.432.S439. (V.V.Nakladem)

Let P be any point in the plane of a triangle ABC. Show that the sumof the squares of the areas of three triangles PBC, PCA and PAB cannotexceed 1

16(PA2 + PB2 + PC2)2.

E561.433.S4310.(Eves)

Given two triangles inscribed in the same circle and such that the Simsonlines with respect to one triangle of the vertices of the other are concurrent(as in E535), prove that the Simson lines with respect to the two trianglesof a point on the common circumcircle are parallel.

E563.433. (N.A.Court)

Let A′, B′, C ′, D′ be the antipodes of the circumcenter O of a tetrahedronABCD on the respective spheres OBCD, OCDA, ODAB, OABC. Showthat the lines AA′, BB′, CC ′, DD′ are generators of a quadric. May thisquadric be a cone ?



Using compass only, construct a regular polygon of 30 sides.

E568.434.S442. (P.D.Thomas)

In a given triangle show that the radical axes of the circumcircle with therepsective circles whose diameters are any three concurrent cevians meet thecorresponding sides in three collinear points.

See also E467.

E569.434.S441. (D.Matlack)

Through a fixed point A on a circle (O), a line is drawn, parallel to a variableradius OP , meeting the circle again at Q. Find the envelope of the chordPQ.

E570.434.S74?,764.(L.M.Kelly)

If the six conics determined by each five of a set of six points are congruent,must they coincide ?

E571.435.S442. (S.Mitchell)

Let O be the midpoint of a chord AB of a circle, and CD, EF any twoother chords through O. Prove synthetically that CE and DF meet AB inpoints equidistant from O.


This is the butterfly theorem. E.P.Starke: This is also true of any conic.Butchart remarks that this is an example of Desargues’ theorem concerningconics passing through the vertices of a complete quadrangle. O is a doublepoint of the involution determined by CDEF on AB. Since A, B form apair of corresponding points in this hyperbolic involution, the other doublepoint is at infinity, and the points G, H (where CE and DF meet ABrespectively) are likewise harmonic conjugates.

E573.435.S442.(N.A.Court)

Given two (three) vertices of a triangle (tetrahedron), determine the remain-ing vertex so that a given point and a given line (plane) shall be harmonicfor the triangle (tetrahedron).

Court refers to AMM 36.p.89. Does harmonic here mean dual?

E574.435.S442. (W.E.Buker)

If a quadrilateral with sides a, b, c, x is inscribed in a semicircle of diameterx, show that

x3 − (a2 + b2 + c2)x− 2abc = 0.


Given an “isosceles” tetrahedron A1A2A3A4 (so that every two oppositeedges are equal), let perpendiculars be drawn to the faces A2A3A4, A3A4A1,A4A1A2, A1A2A3 at their circumcenters O1,O2, O3, O4, to meet the hemi-spheres described exteriorly (or interiorly) on the respective circumcirclesin P1, P2, P3, P4. Show that the tetrahedra O1O2O3O4 and P1P2P3P4 areisosceles, and that they have the same centroid as A1A2A3A4.

E583.437.S444. (N.A.Court)

Given four spheres (A), (B), (C), (D) with centers A, B, C, D, let a planeparallel to ABC cut DA, DB, DC in points U , V , W . Show that the radicalaxix of the three spheres having U , V , W for centers and coaxal with therespective pairs of spheres (D) and (A), (D) and (B), (D) and (C), coincideswith the radical axis of the spheres (A), (B), (C).


E585.437.S74,762. (A.H.Stone)

Let a circle with center O meet the sides BC, CA, AB of triangle ABC inthe pairs of points X and X ′, Y and Y ′, Z and Z ′. Let M be the Miquelpoint, the point of concurrence of circles AY Z, BZX, CXY , and M ′ bethat of X ′Y ′Z ′. Prove that OM = OM ′. Further, if the lines AX,BY,CZconcur, say in P , and consequently the lines AX ′, BY ′, CZ ′ concur, say inP ′, prove that PP ′ and MM ′ are parallel.


Let AA′, BB′, CC ′, DD′ be the altitudes of an orthocentric tetrahedronABCD, with orthocenter H. Show that

BC ·DAB′C ′ ·D′A′ =

CA ·DBC ′A′ ·D′B′ =

AB ·DCA′B′ ·D′C ′ =

HA ·HBHC ′ ·HD′ =

HC ·HDHA′ ·HB′ = · · ·

E591.43?.S446. (C.J.Coe)

Two given coplanar circles, (A1) and (A2), are cut orthogonally by a thirdcircle (B). Prove that a line joining either intersection point on (A1) toeither intersection point on (A2) will pass through one of two points on theline of centers A1A2, these two points being the same for all choices of theorthogonal circle (B).

E593.43?.S445. (N.A.Court)

A variable tetrahedron has three fixed vertices, a fixed circumsphere, andthe sum of the squares of its edges is constant. Find the locus of its Mongepoint.

E595.43?.S446. (H.T.R.Aude)

Find the smallest set of three different integers to represents the sides ofa triangle in which one angle is 60 and each of the other angles differstherefrom by not more than one minutes.

E597.4310.S446. (V.Thebault)

Let P be any point in the plane of a triangle ABC. Let (U), (V ), (W )denote the circles BCP , CAP , ABP , and (U ′), (V ′), (W ′) their images byreflection in the respective sides BC, CA, AB. Also, let u, u′ ne the powers


of A with respect to (U), (U ′), and let v, v′, w, w′ be defined analogously.Show that the circles (U ′), (V ′), (W ′) are concurrent, and that

u+ u′ + v + v′ + w + w′ = a2 + b2 + c2.

E600.4310.S4410. (J.H.Butchart)

If the radii of the fixed and rolling circles are a and b respectively, the lengthof one arch of an epicycloid is 8(a+b)b

a , and the area bounded by one archand the fixed circle is

π(3a2 + 8ab+ 4b2)b2

a(a+ 2b).

Corresponding formulae for the hypocycloid are obtained by changing thesign of b. Prove these formulae synthetically.

E606.441.S448. (L.M.Kelly)

If a set of four coplanar points has the property that the circumcircles of allsubsetsof three are equal (but not coincident), then the set is orthocentric.(See R.A. Johnson, p.75). Establish the existence of an analogous set of fivepoints in space, i.e., such that the circumsphere of all subsets of four areequal (but not coincident).

See Eves’ solutions to E540.


Conside an orthocentric tetrahedron ABCD, of orthocenter H. Let O,A′, B′, C ′, D′ be the circumcenters of the tetrahedra ABCD, BCDH,CDAH, DABH, ABCH. Prove that the tetrahedra ABCD and A′B′C ′D′

are homethetic from a center which divides OH in the ratio 3 : 2. Show alsothat the lines AA′, BB′, CC ′, DD′ passes through the centers of gravity ofthe respectivel tetrahedra BCDH, CDAH, DABH, ABCH.

E609.442.S449. (F.Hawthorne)

Show that the diagonals of three faces of a parallelepiped, drawn from thesame vertex andprolonged half their length, determinethree poitns whichare coplanar with the opposite vertex.


E613.443.S4410. (L.M.Kelly)

Can a triangle have equal symmedians without being isosceles? No, as aconsequence of the following.

Lemma 1: If the internal cevians AX and BY are such that angle BAX > ABY and CAX > CBY , then BY > AX.

If two cevians satisfy the conditions of Lemma 1, so do their isogonalconjugates.

Lemma 2: Two medians satisfy the condition of Lemma 1 unless thetriangle is isosceles.

The famous Steiner - Lehmus theorem is another simple consequence ofLemma 1.


From a given tetrahedron we derive another by taking as vertices the pointsof contact of the insphere with the faces. Show that the dihedral anglesatpairs of opposite edges of the first tetrahedron are supplementary, if andonly if the second tetrahedron is trirectangular.

E619.444.S451. (W.B.Clarke)

Prove that the four triangles of the complete quadrangle formed by thecircumcenters of the four triangles of any complete quadrilateral are similarto those triangles.

E620.444.S451. (A.Wayne)

Find integral sides for a triangle in which one angle is six times another.

E623.445.S452. (N.A.Court)

The circumsphere of a tetrahedron ABCD meets four cevians LA, LB, LC,LD in the points A′, B′, C ′, D′; A′′B′′C ′′D′′ is a tetrahedron homotheticto the tetrahedron A′B′C ′D′ with respect to the point L. If P and Q aretwo points in space, show that the four spheres AA′′PQ, BB′′PQ, CC ′′PQ,DD′′PQ are coaxal.


Construct an enneagon, given the centers of exterior squares described onthe sides.


E633.447.S454. (N.A.Court)

Given a point M and four spheres (A), (B), (C), (D) whose centers forma tetrahedron, let MEE′ be the transversal from M to the two oppositeedges BC, DA, and let spheres (E) and (E′) be constructed coaxal withthe pairs of spheres (B), (C), and (D), (A). We have analogous spheres (F )and (F ′), (G) and (G′). Sho that the sphere (M) coaxal with (E), (E′) islikewise coaxal with (F ), (F ′), and with (G), (G′).

E635.447.S453. (R.A.Rosenbaum)

Derive parametric equations for the involute of the involute . . . (n times) ofa circle (with the same starting point for each processing of unwinding).


Locate a plane which touch four equal spheres inscribed in the trihedra atthe respective vertices of a given tetrahedron.

E639.448.S455. (H.Eves)

A clothoid (or transition spiral, used in highway engineering) is defined asa curve whose curvature varies directly with the arc length. Locate thegeometrical pole of this spiral.

E643.449.S456. (W.E.Beuker)

The sides a, b, c, d of a plane quadrangle being given in order andalso thearea A, find the length of the longer diagonal.

E645.449.S457. (C.D.Olds)

Let P1, P2, P3 be any three points on a plane curve C, and O a point in thesame plane. If the areas of the three triangles OP2P3, OP3P1, OP1P2 areconnected by a relation independent of the coordinates of P1, P2, P3, provethat C is a centric conic and O its center.

E646.4410.S456. (O.Frink)

Prove that any two conjugate planes through a secant of a sphere meet thesphere in orthogonal circles.


E647.4410.S456. (Thebault)

Let (m1,m2,m3,m4) be the barycentric coordinates of a point G with re-spect to a regular tetrahedron A1A2A3A4 of edge a (so that G is the centroidof masses m1 etc at A1). Obtain an expression for the distance A4G.


The ends of a chord UV of the circle r = a have the parametric angles φand kφ, where k is a constant greater than 1. Show that the locus of themidpoint of UV is a prolate epitrochoid whose polar equation is

r = a cosk − 1k + 1

θ.

Show also that the envelope of UV is an epicycloid, the point of contactdividing UV in the ratio 1 : k.

E657.452. (Thebault)

Determine the locus of centers of spheres passing through two given pointsand touching a given sphere.

E659.452.S4510. (R.A.Staal)

Show that, if one conic is self - reciprocal with respect to another, thenthe two conics belong to a symmetrical set of four, each of which is self -reciprocal with respect ot any of the other three. (However, not more thanthree of the four conics can be real).

E661.453. (H.Eves)

A plane p is projected from a point L onto a plane p′. Find those pointson p for which all angles on p having such a point for vertex are invariantunder the projection.


Let P , Q, R, P ′, Q′, R′ be arbitrary points on the respective sides BC,CA, AB, DA, DB, DC of a given tetrahedron ABCD. Prove that the fourplanes, parallel to the aces BCD, CDA, DAB, ABC, drawn through thecentroids of the respective tetrahedra AQRP ′, BRPQ′, PQR′, DP ′Q′R′,form a tetrahedron of constant volume.



Let G be the centroid of n coplanar points Pi, Q any point in the sameplane, and i the signed area of the triangle QGPi. Show that∑

i

i = 0.

E673.455.S461. (L.S.Shively)

Does there exist a regular polygon having both these properties: (1) a di-agonal is equal to the sum of two other diagonals; (2) a diagonal is equal tothe sum of a side and another diagonal.

E675.455.S461. (H.Eves)

Show that the ratio of the curvatures of two curves in a plane at a point ofcontact is invariant under projection.


Let the altitudes of a tetrahedron ABCD meet the circumsphere again in thepoints A′, B′, C ′, D′. Show that the volume of the solid ABCDA′B′C ′D′

is three times that of the given tetrahedron.

E688.458.S465. (P.A.Piza)

Consider the right triangle ABC with sides a = 12, b = 5, c = 13. Keepingthe base BC fixed, displace the vertex A without altering the perimeteruntil the value of cosA is reduced from 5

13 to 738 . Show that the sides of the

new triangle satisfy the relation

a3 = b3 + c3.

E689.458.S465. (M.ward)

Let A, B, C, D be four collinear points in the order written, and let P beany other point in space. Prove that the inequality

PA+ PD ≥ PB + PC

holds for all positions of P if and only if AB = CD.


E693.459.S465. (N.A.Court)

Through a given point to draw a line meeting three planes in three pointsso that the anharmonic ratio of the four points shall have a given value.

E697.4510.S466. (C.A.Murray)

A certain geometry text raises the question whether the following procedurewill inscribe a regular n−gon in a circle: AB being a diameter of the circle,construct an equilateral triagnel ABC. Divide AB into n equal parts andlet D be the second point of division from A. Draw CD, producing it to cutthe circle at E. Is AE the side of a regular n−gon inscribed in the circle?For n = 3, 4 the answer is readily affirmative. Does the procedure yield aregular pentagon for n = 5. If not, give a mesure of the error.

E699.4510.S466. (Thebault)

Let A1, B1, C1; A2, B2, C2; A3, B3, C3 be the feet of the altitudes, thesymmedians, and the cevians through the circumcenter, on teh sides BC,CA, AB of a triangle ABC.

1. The lines B1C1, B2C2, B3C3 are concurrent in a point M1.

2. The lines C1A1, C2A2, C3A3 are concurrent in a point M2,

3. The lines A1B1, A2B2, A3B3 are concurrent in a point M3.

4. The triangle M1M2M3 is homological to, and circumscribes, triangleABC.


Construct a quadrangle given the lengths of its sides and the line joiningthe midpoints of the diagonals.

E709.462.S468. (N.A.Court)

In a given sphere to inscribe a tetrahedron so that three concurrent edgesshall pass through threegiven points and the plane ofthe three remainingedges shall be parallel to the plane determined by the three given points.

A 2-dimensional analogue (W.A.Rees): In a given circle to inscribe atriangle so tha two sidesshall pass through two given points andthe thirdside shall be parallel to the line determined by the two given points.


E711.463.S4610. (H.S.M.Coxeter)

Suppose that the vertices of a polyhedron represent places that we wish tovisit, whilethe edges reprsetn the only possible routes. Hamilton consideredthe problem of visiting all the places, without repetition, on a single journey.This is easily solved for the pentagonal dodecahedron. Prove that it cannotbe done for the rhombic dodecahedron.


Find a euclidean construction for a nonregular pentagon which has both acircumcircle and an incircle.

E715.463.S472. (L.M.Kelly)

Suppose ABCD is a proper plane convex quadrilateral and P a point ex-terior to this plane. Consider the four tetrahedra PABC, PQBD, PACD,PBCD. If PH is the shortest of all the altitudes of these four tetrahedra,show that H must be interior to ABCD.


At the vertices of an equilateral triangle three equal circles are drawn ex-ternally tangent to the circumcircle. Show that one of the three tangents tothese equal circle, from any point whatever on the circumcircle, is equal tothe sum of the other two.


With the vertices A, B, C of an equilateral triangle as centers draw thecircles (A), (B), (C), which are concurrent at the center O of the triangle,and then draw an arbitrary cirlce (D) passing through O. Show that thelength of one of the common tangents to the circles (A) and (D), (B) and(D), (C) and (D) is equal to the sum of the lengths of the other two.

Editorial note: E720 and E728 are both generalizations of the classicrelation MA +MB = MC where M is any point on the minor arc AB ofthe circumcircle of the equilateral triangle ABC. These two problems arespecial cases of Casey’s theorem. See Johnson, pp.121 – 126.


E730.466.S472.(J.H.Butchart)

The interior angle bisectors of a triangle meet the noncorresponding sidesof the medial triangle in six points which lie in pairs on the lines joining thepoints of tangency of the inscribed circle.


Show that the four spheres passing through the Monge point and the ninepoint circles of the faces of a tetrahedron are equal to each other.

E739.468.S474. (L.M.Kelly)

An ellipse inscribed in the triangle ABC is tangent to AB at D. Show thatthe midpoints of CD and AB are collinear with the center of the ellipse.

E741.46p532.S475. (W.Scott)

Prove that in a nondegenerate right spherical triangle with hypotenuse cand legs a, b, we have a2 + b2 > c2.

E742.469.S475. (B.F.Laposky)

Let ABC be a triangle, LMN the median triangle, DEF the orthic tri-angle, O the circumcenter, J the ninepoint center, and T , U , V the otherintersections of the medians AL, BM , CN with the nine-point circle (J).Now, there are two sets of circles tangent to the circumcircle at the verticesA, B, C and also tangent to (J). Show that the circles of one set have theircenters at the intersections of OA, OB, OC with the correpsonding sidesof LMN and touch (J) at D, E, F ; the circles of the other set have theircenters at the intersections of OA, OB, OC with the lines JT , JU , JV andtouch (J) at T , U , V .

E749.4610.S476. (Thebault)

In a given sphere inscribe a right circular cone whoselateral area is equal tothe area of the zone beneath its base. Show that the total area of the coneis equal to the area of the zone in which it is inscribed.



Show that in a right triangle the twelve points of contact of the inscribed andescribed circles form two groups of six points situated on two circles whichcut each other orthogonally at the points of intersection of the cirucmcirclewith the line joining the midpoints of the legs of the triangle.


A kite consists of the area bounded by a major arc of a circle of radius rand the two tangents drawn at the endpoints of the arc. Show that

1. The area of the kite is equalto half the product of its perimeter byrthe radius r.

2. Og : OG = 3 : 2, where g and G are the centroids of he perimeter andarea and O is the center of the kite’s arc.

3. Og′ : OG′ = 4 : 3, wehre g′ and G′ are the centroids of the surface andvolume of the solid of revolution obtained by revolving the kite aboutits axis.

4. The plane through G′ perpendicular to the axis bisects the lateral areaof rthe solid.

5. The volume of the solid is equal to one third the product of its surfaceby the radius r.

E762.473.S479. (J.R.van Andel)

Let A1 and A2 be two circles with radii a1 and a2 and centers (a1, 0) and(a2, 0) respectively, with a2 > a1 > 0.Let C be any circle in the crescentshaped area M between A1 and A2, and tangent to both A1 and A2.

1. The locus of the center of C as its sweeps out M is an ellipse withsemiaxes 1

2(a1 + a2) and√a1a2.

2. If Ct is a circle of radius rt and center Pt(xt, yt), where

rt = a1a2(a2 − a1)φt,xt = a1a2(a2 + a1)φt,yt = 2trt,1φt

= a1a2 + t2(a2 − a1)2,


then, for any real value of t, Ct lies in M and is tangent to A1, A2 andCt−1.


The lines joining the orthocenters of the faces of a tetrahedron to the reflec-tions of the points of intersections of the corresponding altitudes with thecircumpshere, are concurrent at the Monge point of the tetrahedron.

E767.474.S481. (M.Schwartz)

From any point P on BC of parallelogram ABCD line segments are drawnto A and D. from any point Q on AD line segments are drawn to B and C.Through the intersections of these four segments (PA,PD,QB,QC) a lineis drawn meeting AB in R and CD in S. Prove that BR equals DS.


In a plane quadrangle ABCD the perpendicular at A to side AB cuts theopposite side CD at M , and the perpendicular at A to side AD cuts theopposite side BCat N . Show that the radicalaxis ofthe circles described onAM and AN as diameters coincides with the tangent atA to the equilateralhyperbola circumscribing the quadrangle.

E774.474. (N.Anning)

Consider points on the medians of a triangle. Through the centroid nostraight line can be drawn which cut off one-third of the area. Through apoint four-fifths of the distance from vertex to base, four such lines can bedrawn. Find points on the median at which the number of possible lineschanges.


For a given tetrahedron ABCD, find the point P in space such that theshortest parth separating P from each of the vertices A, B, C, D, afterhaving touched the opposite faces BCD, CDA, DAB, ABC are equal toeach other.


E783.477.S484. (C.D.Olds)

Given a parallelogram and its diagonals. Let each side of the parallelogrambe divided into n equal parts and let lines be drawn through the points ofdivision, parallel to the sides and to the diagonals of parallelogram. Findthe total number of triangles in the resulting figure.

E786.478.S485. (M.Goldberg)

Suppose that an equilateral triangle is circumscribed about a regular n−gon,where n = 3k pm1, so that one side of the n−gon lies on one of the sides ofthe triangle. Show that the angle subtended by theis side of the n−gon atthe opposite vertex of the triangle is 2π

3n .


In a triangle,

s4 + (s − a)4 + (s − b)4 + (s − c)4 − a4 − b4 − c4 = 122.


With straightedge alone construct a hexagon which can possess both aninscribed and a circumscribed conic.

E795.479.S488. (N.A.Court)

The pairs of points U ′, U ′′; V ′, V ′′; W ′, W ′′ are marked, respectively, on theedges DA, DB, DC of a tetrahedron ABCD. Determine three points U , V ,W on the edges BC, CA, AB, respectively, so that the three lines joiningthe verticesof each of the triangles DCB, DAC, DBA, ABC, respectively,shall have a common point.

E797.4710.S488. (C.O.Hines)

If ellipses are described on diameters of a given circle as major axis, and suchthat they all pass through a given point (within, or on the boundary of, thecircle), then they will also all pass through a second point, symmetricalabout the center to the first, and the locus of their foci will be an ellipsehaving the two fixed pointsas foci and the common diameter as major axis.



The polar planes, with respect to a tetrahedron, of the isotomic conjugatesof a set of collinear points are coaxal.


If the squares of the sides of a triangle form an arithmetic progression, thenthe line joining the centroid and the symmedian point is parallel to one sideof the triangle.

See also E259 and E411.

E804.482.S548. (S.H.Gould)

Denote by U the ellipsoid a21x

21 + a2x

22 + a3x

23 = 1, by Eb the ellipse of

intersection of U with the plane b1x2 + b2x2 + b3x3 = 0, by (p1, p2, p3) apoint variable on Eb, and bt Ep the ellipse of intersection of U with theplane p1x1 + p2x2 + p3x3 = 0. Determine (p1, p2, p3) so as to minimize themajor axis of Ep.

E805.482.S489. (N.A.Court)

If two coplanar edges of a tetrahedron areeach equal to the respectivelyopposite edge, the remaining two opposite edges are each coplanar with theEuler line of the tetrahedron.

E812.48p248.S491. (M.Dernham)

Find the shortest perimeter common to two different primitive Pythagoreantriangles.

Answer: (364,627,725) and (748,195,773) with common perimeter 1716.See also E18, 67, 73, 283, 324, 327, 380, 828.

E820.485. (K.Tan)

If ABC is an equilateral triangle, and P is any point on the circumference ofthe inscribed circle, prove synthetically that (PA2 +PB2 +PC2 is constant.

E826.48p427.S494. (C.S.Ogilvy)

Find the equation of the ellipse with foci (−1, 0) and (1, 0) and with semi-perimeter equal to the length of one arch of the sine curve y = sinx.



The six planes bisecting the adjacent dihedral angles around the base of atetrahedron, taken four by four, form fifteen tetrahedra circumscribed abouta common paraboloid of revolution.

E831.488.S496. (K.W.Crain)

If squares be constructed on the legs of a right triangle, the lines (which donot lie along the sides of the triangle) drawn from each end of the hypotenuseto a vertex of the opposite square intersect on the altitude which passesthrough the vertex of the right angle.


Inscribe three equal circles (A), B), (C) in the corresponding interior anglesof a triangle ABC such that we may insert between(B) and C) a chain oftangent circles equal to B) and C) and all touching side BC, and similarchains between (C) and (A) and between (A) and (B). What is the conditionfor possibility of solution, and how many solutions are there for a giventriangle ABC?

E847.491.S497(corrected). (A.Newhouse)

Let a, b, A be the given parts of a triangle in the ambiguous case. Showthat the area of the triangle is given by

K =b

sinA[b cosA±

√a2 − b2 sin2A].

E849.491.S498. (Trigg)

The area of a triangle is to the area of the triangle determined by the pointsof contact of its incircle (or excircle) as its circumdiameter is to its inradius(or exradius).


The area of a quadrilateral which has both a circumcircle and an incircle isequal to the square root of the product of its sides.



Planes through the orthocenter of an orthocentric tetrahedron perpendicularto four coplanar cevians cut the spheres described on these cevians in fourcospherical circles.

E859.493.S4910. (C.W.Trigg)

If the faces of a hexahedron are equilateral triangles congruent to the facesof a regular octhedron, then the radii of the inscribed spheresare in the ratio2:3.

E862.494. (R.E.Horton)

Find the rectangle of greatest and least area which can be circumscribedabout a given parallelogram.


Find a point such that planes drawn through this point parallel to the facesof a tetrahedron cut the opposite trihedrals in equivalent triangles. Expressthe common area of these triangles in terms of the areas of the faces of thetetrahedron.

E868.495. (P.D.Thomas)

Let P and Q be, respectively, the feet of the common perpendicular to twofixed skew lines p and q. A variable line r meets p in R and q in S. Findthe locus of r if the volume of the tetrahedron PQRS is constant. Also findthe locus of the centroid of PQRS.

E870.495. (J.Rosenbaum)

Characterize quadrilaterals A1B1C1D1 such that if A2, B2, C2, D2 are thecircumcenters of A1B1C1, B1C1D1, C1D1A1, D1A1B1, then A1, B1, C1, D1

are the circumcenters of A2B2C2, B2C2D2, C2D2A2, D2A2B2.

E878.497. (K.Tan)

Let S be the incenter of the right triangle ABC, and X the point of contactof the hypotenuse BC with the incircle. With center X and radius XSdescribe the circle cutting BS, CS at M and N respectively. Let AD be the


altitude on BC. Show that M and N are the incenters of the right trianglesABD and ACD respectively.

E879.497. (J.Langr)

Let S1, S2, S3 be the midpoints of three concurrent cevians of triangle ABC.Let S2S3, S3S1, S1S2 meet the sides BC, CA, AB in A1, B1, C1; A2, B2,C2; A3, B3, C3 respectively. Show that

1. A2, A3; B3, B1; C1, C2 are isotomic points on the segments BC, CA,AB;

2. A1, B2,C3 are collinear;

3. A2, A3, B3, B1, C1, C2 lie on a conic.

E882.498.(Trigg)

Five regular tetrahedra arranged around a common edge just fail to fill spacearound that the edge.

(a) What is the closest approximation to a regular tetrahedron suchthat five such tetrahedra will fill the space around a common edge to forma decahedron having equilateral faces ?

(b) Show that the edge of the decahedron is√

5 times the radius of thesphere touching those edges which radiate from the axis of the decahedron.


In the tetrahedron ABCD let A1, B1, C1, D1 divide a set of concurrentcevians AA′, BB′, CC ′, DD′ in the same ratio, and let A1B1, A1C1,, A1D1

pierce BCD in A2, A3, A4. Show that triangles A2CD, A3DB, A4BC haveequal areas.

E892.4910. (J.P.Ballantine and G.E.Ulrich)

Let T be a given triangle, U the triangle whose vertices are the centroids ofequilateral triangles described externally onthe sides of T , and V the trianglewhose vertices are the centroids of equilateral triangles described internallyon the sides of T . Show that area T = area U - area V .



Let perpendiculars through vertex A of tetrahedron ABCD to the facesABC, ACD, ADB cut the circumsphere of ABCD in B′, C ′, D′ respec-tively. Show that the volumes of the polyhedraA−B′C ′D′−B, A−B′C ′D′−C, A−B′C ′D′ −D are equal to that of ABCD.

E895.4910. (L.Fejes Toth)

Let the incircle of a convex polygon be defined as the largest circle whoseinterior lies in the interior of the polygon. Show that the sum of the squaresof the edges of a convex polyhedron isat least twelve times the squares ofthe diameter of the least incircle of the faces.

E1073.417.S751,762.(G.W.Walker)

A polygonal spiral A1A2A3 · · · of unit segments winds counter- clockwiseand is constructed in the following manner: Point A1 is at the origin, pointA2 is at (1, 0), An−1AnAn+1 = 2π

n for all n ≥ 2. Is there a point lyingwithin the interior of each An−1AnAn+1 ? If so, what are its coordinates ?


American Mathematical Monthly

Articles

V.O.McBrien, Cardioids associatedwith a cyclic quadrangle, 51 (1944) no.2,pp.74 – 77.

Douglas Derry, Affine geometry of convex quartics, 51 (1944) no.2, pp.78– 83.

L.Brand, The eight-point circle and the nine-point circle, 51 (1944) no.2,pp.84–85.

O.S.Adams, Notes on the geometry of the triangle, 51 (1944) no.2, pp.85 – 87.

H.Eves, Concerning some perspective triangles, 41 (1944) no.6, 324-331.Eves, H., Feuerbach’s Theorem by Mean Position, 52 (1945) no.1, pp.35–

36.Frame, J.S., An Approach to the normal curve and the cycloid, 52 (1945)

no.5, pp.266 – 269.Sandham, H.F., A simple proof of Feuerbach’s theorem, 52 (1945) no.10,

p.571.J.H.Butchart, Some properties on the Limacon and cardioid, 52 (1945)

no.7, pp.384 – 387.Mordell, L.J. Rational points on cubic curves and surfaces, 51 (1945)

332 – 339.Thebault, V., A theorem concerning circles, 53 (1946) no.1, pp.27 – 28.Thebault, V., Some spheres associated with a tetrahedron, 53 (1946)

p.89.**H.E.Fettis, The Fermat and Hessian points ofa triangle, 53 (1946)

no.2, pp 74–78.H.S.M.Coxeter, Quaternions and reflections 53 (1946) no.3, pp.136 – 146;

postscript, p.588.R. Goormaghtigh, Pairs of triangles inscribed in a circle 53 (1946) no.4,

pp.200 – 204.S.M.Karmelkar, Construction of the in-Feuerbach point, 53 (1946) no.4,

pp.206–207.P.A.Piza, Elliptic Fermagoric triangles, 53 (1946) no.6, pp.317–323.V.Thebault, Concerning pedal circles and spheres, 53 (1946) no.6, pp.324

– 326.C.E.Springer, Volume coordinates, 53 (1946) no.7, pp.377 – 382.


R.Goormaghtigh, On the Feuerbach points, 53 (1946), no.8, p.453.M.L.MacQueen and R.W.Hartley, Elliptic Euleroid, 53 (1946) no.9, pp.511

– 516.A.E.Hatleman, A geometric approach to the covariants of a cubic, 53

(1946) no.9, pp.517 – 520.E.Mitchell, Conjugo - Conjugate couples in involution, 54 (1947) no.1,

pp.16 – 23.R.Goormaghtigh, Orthopolar and isopolar lines in the cyclic quadrilat-

eral, 54 (1947), no.4, pp.211–214.H.F.Sandham, An approximate construction for e, 54 (1947) no.4, pp.215–

216.V.Thebault, Nagel point in the tetrahedron, 54 (1947) no.6, pp.275 –

276.R.Goormaghtigh, The Hervey point of the general n−line, 54 (1947),

no.6, pp.327–331.V.Thebault, Concerning the Euler line of a triangle, 54 (1947) no.8,

pp.447 – 452.V.Thebault, Tetrahedrons having a common face, 54 (1947) no.7, pp.395–

398.G.T.Williams and D.H.Browne, A family of integers and a theorem on

circles, 54 (1947) no.9, pp.534 – 536.V.Thebault, Theorem on trapezoid, 54 (1947) no.9, pp.537 – 538.**L.Droussent, On a theorem of J. Griffiths, 54 (1947) no.9, pp.538 –

540. 4

**C.E.Noble, An anallagmatic cubic, 55 (1948) no.1, pp.7 – 14.R.Goormaghtigh, On the two-angle pole of a line to a triangle, 55 (1948),

no.2, pp.71 – 75.J.H.Butchart, Rotation of the tangent to a hypocycloid, 55 (1948) no.3,

pp.152 – 153.F.H.Young, The ellipse as a circle with a moving center, 55 (1948) no.3,

pp.156–158.W.H.Bunch, The quadrilateral of Pascal’s hexagram, 55 (1948) no.4,

pp.210 – 217.N.A.Court, Notes on cospherical points, 55 (1948) no.4, pp.218 – 221.T.Ladopoulos, Some theorems on cyclic polygons inscribed in a circle,

55 (1948) no.5, pp.301 – 307.4Here, the definition of the Steiner ellipse is recalled: it is the inscribed ellipse which

touches the sides of the triangle at their midpoints.


V.Thebault, On the twelve point sphere of the tetrahedron, 55 (1948)no.6, p.357.

Shou Chen, On the application of vector algerbra to projective geometry,55 (1948) no.9, pp.541 – 544.

A.Wormser, Polygons with two equiangular points, 55 (1948) no.10,pp.619 – 629.

**R.Goormaghtigh, On anallagmatic cubics, 55 (1948) no.10, p636.V.Thebault, On the altitudes of the triangles and of the tetrahedron, 55

(1948) pp.637 – 638.


American Mathematical Monthly,

Problems before 1940

AMM2892.S227. (R.T. McGregor)

Two parabolas have parallel axes. Prove that their common chord bisectstheir common tangent.

AMM2933.21:467S23:147–148. (H.E. Dudeney)

Dudeney’s Problem (1902): With ruler and compasses only, divide an equi-lateral triangle into four rectilinear pieces which may be put together so asto form a square. 5

AMM2994.S242. (R.M. Matthews)

Can the following construction be made without the use of regulus? Con-struct a line which meets four given skew lines.

AMM3089.247.S291. (N. Anning)

Given four points, O, A, B, C, on a straight line, to construct, with straight-edge only, the point P on the line such that OP shall be the harmonic meanof OA, OB, OC.

Solution by Otto Dunkel: the intersection of the line with the dual of Owith respect to any triangle bounded by three lines each through A, B, andC.

AMM3092.S255. (N.A. Court)

What must be the relations between the coefficients of a cubic equation inorder that its roots, considered as lengths, shall form a triangle? Solution.Let p, q, r be positive numbers such that the roots of the equation x3 −

5Solution by H.C. Bradley. Otto Dunkel’s remark: Dudeney gave without proof inCanterbury Puzzles a solution which he presented to the Royal Society.


px2 + qx− r = 0 are the sides of a triangle. Since these roots are real, thediscriminant

18pqr − 4p3r + p2q2 − 4q3 − 27r2

should be positive. Let s := 12(a+b+c) be the semiperimeter of the triangle.

By the triangle inequality, s − a, s − b, s − c should all be positive. Theseare the roots of the cubic equation

8y3 − 4py2 + (8q − 2p2)y − (4pq − p3 − 8r) = 0.

We require these coeffcients to be positive. Now, 8r < p(4q − p2) is enoughto guarantee these coefficients to be all positive. Thus,

4pq > p3 + 8r,pq(pq + 18r) > 4p3r + 4q3 + 27r2

are necessary and sufficient conditions on the positive numbers p, q, r toguarantee that the roots of the equation x3−px2 +qx−r = 0 form the sidesof a triangle.

AMM3137.S266. (H. Langman)

Show how to draw, using straightedge only, a tangent to the circumference(or an arc) of a circle at a given point, without making use of Pascal’shexagon theorem.

AMM3207.S277. (C.N. Mills)

Prove that 14m

2√

3 is the maximum area of a triangle which can be formedwith the lines a, b, c subject to the condition that a3 + b3 + c3 = 3m3.

AMM3296.S291. (J. Rosenbaum)

It is well known that the radius of the inscribed circle of a right triangle isequal to half the difference between the sum of the legs and the hypotenuse.Derive an analogous expression for the radius of the inscribed sphere of aright tetrahedron.


AMM3367.S302. (H. Langman)

Given any triangle. On each side construct an equilateral triangle exter-nally. The centers of these triangles determine another equilateral triangleA. Similarly an equilateral triangle B is determined by constructing theequilateral triangle internally. Show that the difference between the areasof the triangles A and B is equal to the area of the given triangle.

AMM3421.S313. (O. Dunkel)

A convex polygon of n sides may be divided into triangles by its diago-nals which intersect only at their extremities. Derive an expression for thenumber of ways in which this may be done.

AMM3440.S313. (A. Pelletier)

A triangle is circumscribed about a circle. Prove that the following threelines are concurrent:

1. the line joining the points of contact of any two sides;

2. the line joining the points of intersection of these sides with the bisec-tors of the opposite angles;

3. the line joining the feet of the altitudes on these sides.

AMM3565.S336. (O. Frink)

Find the ellipse of least area circumscribing a given triangle.

AMM3586.S339,349. (R.E.Gaines)

If while an ellipse is turned about in its plane it remains tangent to a fixedstraight line at a fixed point, its foci trace a curve whose area is 2πa(a− b).

AMM3594.S341. (H.T.R.Aude)

Find sets of integers for rational right triangles which, as the number in-crease, approach a 30 − 60 right triangle.


AMM3658.S354. (J.M.Feld)

The Simson line of a point P on the circumcircle of a triangle ABC is thetangent at the vertex of a parabola tangent to the sides of ABC and havingits focus at P .

AMM3683.S3510. (R.Robinson)

Show that the sum of the medians of a simplex of n dimensions is smallerthan 2

n and greater than n+1n2 times the sum of the edges of the simplex, and

that these are the best limits that can be given.

AMM3713.S (R.E.Gaines)

Determine the position of a normal chord of a conic which forms a segmentof minimum area. Find the area of such a segment of an ellipse.

AMM3718. (F.Morley)

Show that the ellipse through the points given by the complex numbers a,b, c and with center 1

3(a+ b+ c) has semi-axes whose lengths are

13

(|a+ ω2b+ ωc| ± |a+ ωb+ ω2c|

),

where ω = 12(−1 + i

√3).

AMM3726. (M.Charosh)

The vertices of a triangle inscribed in a given circle are the points of tangencyof a triangle circumscribed about the circle. Prove that the product of theperpendiculars from any point on the circle to the sides of the inscribedtriangle is equal to the product of the perpendiculars from the same pointto the sides of the circumscribed triangle.

AMM3740. (Erdos)

From a point O insdie a given triangle ABC the perpendiculars OP , OQ,OR are drawn to its sides. Prove that

OA+OB +OC ≥ 2(OP +OQ+OR).

This is the famous Erdos-Mordell inequality. See also AMM 2001(2)165–168, S. Dar and S. Gueron, A weighted Erdos-Mordell inequality.


AMM3743. (N.Anning)

Two congruent coplanar parabolas have the same line as axis and open inthe same direction. Tangents are drawn to the inner from any point of theouter. Prove that the area bounded by the tangents and the arc joiningtheir points of contact is invariant.

AMM3746. (Erdos)

Given a triangle ABC, with the sides a > b > c, and any point O in itsinterior. Let AO, BO, CO cut the opposite sides in P , Q, R. Prove that

OP +OQ+OR < a.

AMM3763. (Erdos)

Given any simple polygon P which is not convex, draw the smallest convexpolygon P ′ which cntains P . This convex polygon P ′ wil contain the areaP and certain additional areas.Reflect each of these additional areas withrespect to the corresponding added side, thus obtaining a new polygon P1.If P1 is not convex, repeat the process, obtaining a polygon P2. Prove thatafter a finite number of such steps a polygon Pn will be obtained which beconvex.

AMM3776. (E.P.Starke)

Determine all triangles whose sides are relatively prime integers and suchthat one angle is double another.

AMM3780. (J.M.Feld)

In triangle A1A2A3 the transversal AiDi divides AjAk in the ration AjDi :DiAk = pi : qi, where ijk is a cyclic permutation of 123. The transversalAiDi and AjDj intersect in Pk. Find the value of the cross ratio

P3P2P2A1

P3D1D1A1

in terms of the p’s and q’s. Show that Ceva’s theorem is a special case.


AMM3797.?.S38.483.

AMM3805. (R.E.Gaines)

Determine a point on b2x2 − a2y2 = a2b2 such that the tangent and normallines at that point shall be normal and tangent respectively to b2x2−a2y2 =−a2b2, and hence, that if the hyperbola and its conjugate be together con-sidered as a single curve, (b2x2 − a2y2)2 = a4b4, a rectangle may be drawnwhich is both an inscribed and a circumscribed figure.

AMM3831.?.S39.454.

AMM3838.37p395.S?. (J.R. Musselman)

If I is the incenter of triangle A1A2A3 with Ii the point of contact on theside AjAk, and I ′i the image of I in the side IjIk, then the circles I ′iI ′jIk passthrough φ, the point of Feuerbach of A1A2A3, also the circles IiIjI ′k meeton I ′1I ′2I ′3 at a point ψ such that φ and ψ are symmetric as to the center ofthe common nine-point circle of I1I2I3 and I ′1I ′2I ′3.

AMM3839.39p604.S.

Also AMM3991.413.S428. (V.Thebault).

AMM3840.37?.S407. (V.Thebault)

Solid geometry.

AMM3848.37?.S408. (Erdos)

Let O be an arbitrary point in the interior of triangle ABC, and let A′, B′,C ′ be the points in which AO cuts BC etc. IF AA′ ≥ BB′, and AA′ ≥ CC ′,then AA′ ≥ OA′ + OB′ + OC ′, where equality holds only if AA′ = BB′ =CC ′.


Let O be the circumcenter of the tetrahedron ABCD, and P an arbitrarypoint in space. The segments of straight lines PA, PB, PC, PD are dividedin the same ratio u, and the points of division are taken as centers of fourspheres with radii v · PA, v · PB, v · PC, vPD respectively. The radical


center R of these four spheres is on the straight line OP so that

OR : OP = (v2 − u2 + u) : u.

This is a generlization of a proposition by N.A.COurt in which P ≡ G. SeeAMM, 1932, p.198.

AMM3850.37?.(corrected in) AMM3921.39?.S414. (V.Thebault)

Let BCA1A2, CAB1B2, ABC1C2 be squares constructed interiorly on thesides of a triangle ABC for which V is the angle of Brocard. If cotV = 2,the lines which join A, B, C respectively, to the symmetrics of A1, B1, C1

with respect to A2, B2, C2 meet in a point.

AMM3853.38?.S401. (N.A.Court)

Two tangent spheres (A), (D) are each touched by the spheres (B) and (C).The two lines joining an arbitrary point of (A) to the points of contact ofthis sphere with (B) and (C) meet the latter two spheres again in two pointscoplanar with the two points of contact of (D) with (B) and (C).

AMM3857.381.S403. (V.Thebault)

In a triangle the minor auxiliary circle of the Brocard ellipse is tangent tothe nine-point circle. (Note: the Brocard ellipse for a triangle is tangent toits sides and has for its foci the Brocard points of the triangle).

See also AMM4043.Goormaghtigh: This is a special case of a well known generalization of

Feuerbach’s theorem: if two isogonal conjugates and the circumcenter arecollinear, then their common pedal circle touches the ninepoint circle at theorthopole of the circumdiameter containing these points.

Thebault: If a conic is inscribed in a triangle so that one of its principalaxes passes through the circumcenter of the triangle, the auxiliary circle forthat axis is tangent to the nine-point circle.

AMM3860.382.S413.(J.Rosenbaum)

Given a tetrahedron, find the point such that the sum of its distances fromthe vertices is minimum.



A convex polygon A1A2 · · ·A2n with its opposite sides equal and parallel, isinscribed in a conic. IF from a point of the conic parallels are drawn to thedirections conjugate to those of the sides, the intersections these parallelswith the conjugates sides form a polygon of constant area, and its oppositesides have conjugate directions.


Through the vertices of a triangle ABC perpendiculars to its plane aredrawn

AA1 = −AA2 = BC, BB1 = −BB2 = CA, CC−1 = −CC2 = AB.

Show that the points A1, B1, C1, A2, B2, C2 and the vertices of the triangleanticomplementary to ABC lie on the same sphere.

AMM3866.383.S404. (J.M.Feld)

Show that the equation of the circumcircle of the triangle whose sides whavethe equations Li ≡ aix+ biy+ ci = 0, i = 1, 2, 3, can be written in the form

(a21 + b21)(a2b3)L2L3 + (a2

2 + b22)(a3b1)L3L1 + (a23 + b23)(a1b2)L1L2 = 0,

where (aibj) := aibj − ajbi.

AMM3867.383.S404.(V.Thebault)

Given a hexagon A1A2A3A4A5A6 whose consecutive sides are perpendicu-lar, show that the diagonals A1A4, A2A5, A3A6 meet in a point M . LetD1, D4, D2, and D5, D3, D6 be the perpendiculars to A1A4, A2A5, A3A6

at the corresponding extremities; and let B1, B2, B3, B4, B5, B6 be theintersections of (D6,D1), (D1,D2), (D2,D3), (D3,D4), (D4,D5), (D5,D6).The hexagon B1B2B3B4B5B6 is inscriptible in a circle through M . (3) Theareas of the A and B hexagons are equal.

AMM3869.383.S405. (O.Dunkel)

Two perpendicular planes are tangent to a parabolid and their intersectionhas a given direction. Determine the form of the surface generated by the in-tersection as the pair of planes varies. This supplements AMM3773.362.S382.



Let O, I, Ia, Ib, Ic be the centers of the circumcircle, incircles, and ex-circlesof a given triangle ABC; and let Qa, Qb, Qc, Q′

a, Q′b, Q

′c be the intersections

of the sides of the triangle with the interior and exterior bisectors of itsangles. The parallels to the Euler line through the orthogonal projectionson Q′

aQ′bQ

′c, QbQc, QcQa, QaQb of an arbitrary chosen point of that line

meet respectively OI, OIa, OIb, OIc in M , Ma, Mb, Mc. SHow that thesum of the powers of O with respect to the circles with diameters IM , IAMa,IbMb, IcMc is zero.


If a conic is inscribed in a triangle so that one of its principal axes passesthrough the circumcenter of the the triangle, the auxiliary circle of the coniccorresponding to the axis considered is tangent to the nine-point circle ofthe triangle, and conversely.


A convex quadrilateral is circumscribed about a circle. Show that thereexists a straight line segment with ends on opposite sides dividing both thepermieter and the area into two equal parts. Show that the straight linepasses through the center of the inscribed. Consider the converse.

AMM3882.38?.S406. (J.R.Musselman)

The pedal circle of the centroid G of a triangle A1A2A3 passes through thecenters of the hyperbolas of Kiepert and Jerabek. These hyperbolas areconsidered in Casey’s Analytic Geometry of the Point, Line, and Circle,pp.442–448.


The orthopole of the Euler line of a triangle A1A2A3 as to the same triangleis the center of the hyperbola of Jerabak; the orthopole of the Brocard line ofthe triangle as to the same triangle is the center of the hyperbola of Kiepert.(Orthopole is defined in Johnson’s Modern Geometry, p.247.)


AMM3884.38?.S406. (Coxeter)

Prove that the points of contact of real bitangents to a plane quartic ofgenus 3 are its points of intersection with 1,2,4 or 7 conics.

AMM3886.38?.S406. (V.Thebault); special case of AMM3861.402.

A parallelogram is inscribed in an ellipse and a point P is chosen arbitrarilyon the ellipse. Two straight lines are drawn from P parallel to the sidesof the parallelogram cutting them in four points. A third straight line isdrawn from P parallel to one of the diagonals of the parallelogram cuttingthe tangents at the ends of this diagonal in two points. Show that the sixpoints thus obtained are the vertices of a hexagon whose consecutive sidesare parallel to two conjugate diameters of the ellipse, and that the area ofthe hexagon is the same as that of the parallelogram.

AMM3887.38?.S837,8710.(Thebault)

Let P be a quadrilateral inscribed in a circle (O) and let Q be the quadri-lateral formed by the centers of the four circles internally touching (O) andeach of the two diagonals of P . Then the incenters of the four triangleshaving for sides the sides and diagonals of P form a rectangle inscribed inQ.

See editorial comment on 837.p486.Restatement Let P be a quadrilateral inscribed in a circle (O) and let

Q be the quadrilateral formed by the centers of the four circles internallytouching (O) and each of the two diagonals of P . Then the incenters ofthe four triangles having for sides the sides and the diagonals of P form arectangle R inscribed in Q.

AMM3888.38?.S406. (R.Goormaghtigh)

solid geometry


A triangle ABC is inscribed in a circle (O) with a fixed diameter D, and atransversal D′, which turns about a fixed point, cut BC, CA, AB in A1, B1,C1. Let A2 and A3, B2 and B3, C2 and C3 be the orthogonal projections ofA and A1, B and B1, C and C1 on D.


(1) Show that the circles with centers at the midpoints of AA1, BB1,CC1 and passing through A2 and A3, and B2 and B3, and C2 and C3 meetin a fixed point on the Euler circle of the triangle.

(2) Find the locus of the second point of intersection of the three circles.

AMM3890.38p554.S434. (V.Thebault)

Given four straight lines 1, 2, 3, 4 in a plane; through the orthog-onal projections of the vertices of the triangles T1 ≡ (2,3,4), T2 ≡(1,3,4), T3 ≡ (1,2,4), T4 ≡ (1,2,3) on 1,2,3,4 re-spectively, parallels are drawn to the sides opposite the corresponding ver-tices of the triangle considered: these parallels determine four other trianglesT ′

1,′2, T

′3. T

′4 symmetrically equal to the first.(1) Show that the Miquel circles of the quadrilaterals (T ′

1,1), (T ′2,2),

(T ′3,3), (T ′

4,4) are equal to the nine-point circles of T1, T2, T3, T4 andtangent to the circles of T ′

1, T′2, T

′3, T

′4. (2) Show that the Miquel points of

the above quadrilaterals are collinear.


The Apollonian circle of a triangle meet in two points, the Hessian pointsh1 and h2. Show that the two Beltrami points (inverse in the circumcircle ofthe Brocard points) form with either Hessian point an equilateral triangle.Hence, each Beltrami point is the center of the circle passign through h1 andh2 and the other Beltrami point. Naturally, a Brocard point lies on eachcircle.

AMM3893.38?.S409. (N.Anning)

From the vertices of a regular n−gon three are chosen to be the vertices ofa triangle. Prove that the number of essentially different possible trianlgesis the integer nearest to n2

12 .

AMM3895.38p631.S477. (Thebault)

AMM3896.38?.S409. (W.B.Clarke)

Let P and Q be isotomic conjugate points with respect to triangle ABC.Find the locus of P if PQ is parallel to a side of the given triangle.


AMM3897.39p696(corrected).S423. (V.Thebault)

Let ABCD be a rectangle inscribed in a circle with center O, and P a pointon the equilateral hyperbola circumscribing ABCD. The straight line PA,PB, PC, PD cut the circle again in A′, B′, C ′, D′. The perpendicualrsfrom P to the sides of the quadrilateral A′B′C ′D′ cut A′B′ in A′′, B′C ′ inB′′ etc.

(1”) The diagonals A′′C ′′ and B′′D′′ are perpendicular and intersect in apoint Q on the straight line OP .

(2”) The ratio of the lengths of these diagonals is the same as the ratio ofthe sides of the rectangle.

(3”) The quadrilateral A′′B′′C ′′D′′ is inscribed in a circle and circumscribesa conic with foci P and Q.

(4”) The Newton line for A′B′C ′D′ for A′B′C ′D′passes through P and isperpendicular to the Newton line for A′′B′′C ′′D′′.

AMM3898.38?.S4010. (O.Dunkel)

A point is chosen on a rectangular hyperbola. In how many ways, and underwhat conditions, may two other points on the curve be selected so that thecentroid of the three points will lie also on the curve ?


Given the positive integers a, b, c such that a2 = b2+c2, the positive numbersm and n may be determined so that (1) a = m+ n+

√2mn. Conversely, if

2mn is a perfect square, then a in (1) is such that its square is the sum oftwo squares. Show that if x = 2(m+n)

mn , the numbers

(bx+ 1)2 − 1, (cx+ 1)2 − 1, (ax− 1)2 − 1

are also the sides of a right triangle, the last term being the length of thehypotenuse. Determine the smallest integral value of x so that the sidesmay be expressed as integers.


A point M is chosen arbitrarily on the circumcircle of the triangle ABC,and th chords MA′, MB′, MC ′ are drawn parallel to BC, CA, AB. Show


that the orthopoles of the circumcircle diameters through A′, B′, C ′ are thevertices of a triangle equal to th e orthic triangle of ABC. Generalize.

AMM3904.392.S411.(R.P.Baker)

ABC is a given triangle; find the condition that a point P may be con-structed in the plane of ABC such that

PA : PB : PC = p : q : r.

See also PME825.94S.

AMM3911.39?.S411.(J.R.Musselman)

If N be the center of the nine-point circle of ABC, and L′, M ′, N ′ be thesymmetrics of A, B, C respectively as to N , show that the circles AM ′N ′,BN ′L′ and CL′M ′ meet on the circumcircle of ABC at Φ, the point ofFeuerbach for the tangential triangle of, the circumcircle of ABC at itsvertices. ABC, that is the triangle formed by the tangents to


If G be the centroid of the triangle ABC, and L′, M ′, N ′ be the symmetricsof A, B, C respectively as to G, show that the circles AM ′N ′, BN ′L′ andCL′M ′ meet on the circumcircle of ABC at the Steiner point of ABC.

AMM3921.39?.S414.(V.Thebault) = corrected form of AMM3850.37?.S401.

Let BCA1A2, CAB1B2 and ABC1C2 be similar rectangles constructed uponthe sides BC = a, CA = b, AB = c of triangle ABC, of area S, the threerectangles being all directed interiorly or all exteriorly, and CA1

a = AB1b =

BC1c = k. Let Ah, Bh, Ch be points on A1A2, B1B2, C1C2 such that

A1AhA1A2

= B1BhB1B2

= C1ChC1C2

= λ. The straight lines ABh, BCh, CAh determine atriangle similar to ABC, of area

(k cot V − λ)2

k2 + λ2S,

where V is the angle of Brocard for ABC.


AMM3922.39?.S414.(V.Thebault)

The triangle BAC is right angled at A; the squares CAA1C1, ABB1A2 arecaonstructed exteriorly on the sides CA and AB; and M is the foot on BCof the exterior bisector of angle A.

(1) Show that the polygon P ′, the antipedal of M with respect to thepolygon P = BB1A2CC1A1, is inscribed in the circle Σ, passing throughM and concentric with the square constructed interiorly on the hypotenuseBC.

(2) Express the radius r of Σ as a function of elements of triangle BAC,and obtain the condition that r = BC.

(3) Show that the areas of P and P ′ are equal.Note: The antipedal triangle of a point M with respect to a triangle

ABC is determined by the intersection of the perpendiculars to MB andMC, ti MC and MA, and to MA and MB at the respective points B, A,C.

AMM3923.39?.S414.(R.E.Gaines)

It is known that the circumcircle of the triangle formed by three tangentsto a parabola passes through the focus. Show that the diameter d of thecircle is given by d sinα sin β sin γ = a, where α, β, γ are the angles whichthe tangents make with the axis of the parabola y2 = 4ax.


The triangle ABC is right angled at A. The parallel to BC through Ia cutsAB in N and AC in M ; the orthogonal projections of M and N on BC areP and Q. Show that

(1) MQ−MN = r, MQ−MP = BC2

2r ;(2) the circumcircle of rectangle MPQN is tangent to each of the excir-

cles;(3) if D and E are the other intersections of this circle with AB and AC,

then MP = DE = QN = ra, and the lines MP , DE, QN are tangent to acircle concentric with this.


If O is the circumcenter of A1A2A3, and Bi is the image of Ai in the sideAjAk, show that the circles AiOBi, i = 1, 2, 3, meet in that point which is


the inverse in the circumcircle of the isogonal conjugate point of the nine-point center.


The perpendiculars to the sides of triangleA1A2A3 from any point T onthe circumcircle of the triangle cut the circle again in the points B1, B2, B3.Show that the image lines of Bi cut the sides AjAk in three collinear points.The line of these points is ∆2 of Problem 3758, (1937,p.668)..


Let G be the centroid of the tetrahedron ABCD. Through A, B, C, Dplanes are drawn perpendicular to GA, GB, GC, GD respectively, formingthe antipedal tetrahedron of G, with respect to ABCD, of volume Vg. Sim-ilarly, Va, Vb, Vc, Vd are the volumes of the antipedal tetrahedra of A, B, C,D, with respect to the tetrahedra GBCD, GCDA, GDAB, GABC. Showthat Vg = 4Va = 4Vb = 4V − c = 4V d.

AMM3936.39?.S415. (N.A.Court)

If of the four circles determined by four coplanar points taken three at atime two circles are orthogonal, the remaining two circles are orthogonal.(Mathesis, 1929, p.130, Q 2515).

If of the five spheres determined by five points in space taken four at atime three spheres are mutually orthogonal, the remaining two spheres areorthogonal to each other. Prove, or disprove.



Advanced Problems, 1940 – 1949


A given circle has the fixed chord BC and a varibale point A on its circum-ference; the midpoints of CA, AB are B1, C1; the centers of the inscribedand escribed circle for the angle A of triangle ABC are I and Ia; the par-allels to AB through I, Ia meet AC in M , M ′; and the parallels to ACthrough I, Ia meet AB in N , N ′. Prove that

(1) the altitudes of AB1C1 from B1 and C1 pass each through a fixedpoint;

(2) the circles tangent to the sides of angle A with centers at the ortho-centers of triangles IMN , IaM ′N ′ envelop a fixed circle;

(3) the locus of the midpoints of MN and M ′N ′ is a limacon of Pascal.

AMM3938.401.S416. (N.A.Court)

Solid geometry.

AMM3941.402.S416. (N.A.Court)

The polar plane of a point common to three given spheres, with non-collinearcenters, with respect to a varaibel sphere tangent externally to the threegiven spheres, describes a coaxal pencil.


If in a triangle the distances d1, d2, d3 of the midpoints of the sides of thetriangle to a tangent to the nine-point circle satisfy a relation

√d1 ±

√d2 ±√

d3 = 0, the point of contact of that tangent is one of the Feuerbach points.Solution by R.Goormaghtigh. According to Godt’s theorem (Munchener

Berichte, 1896), the Feuerbach points of a triangles are the four points onthe nine-point circle such that the distance of any of them to the midpointof one of the sides equals the sum of its distances to the midpoints of thetwo other sides.


But, if d1 is the distance of the midpoint Am of the side BC of a triangleABC to the tangent at a point T to the nine-poitn circle, then

d1 =TA2

m

R,

R being the circumradius of ABC. The proposed theorem is thereforeanother form of Godt’s theorem.

AMM3946.402. (V.Thebault)

solid geometry.

AMM3947.403.S417. (N.A.Court)

If M , M ′ are two isogonal conjugate points for the tetrahedron ABCD, Sthe projection of M upon the plane ABC, and ′ the point common to theplanes perpendicualr to the lines M ′A, M ′B, M ′C at the points A, B, C,show that the line SS′ and its three analogs PP ′, QQ′, RR′ have a point incommon.

AMM3952.40?.S418. (N.A.Court)

AMM3953.40?.S418. (N.A.Court)


A triangle with unequal sides has one angle of 60 or 120, and a sideadjacent to that angle of length m, a prime.

(1) Determine the lengths of the other two sides so that they are integers.(2) Show that to each value of m there correspond two triangles such

that the difference of their perimeters is a perfect square in one case, andin the otehr case the sum of the perimeters increased by unity is the sum ofsquares of two consecutive integers.

AMM3956.404.S.4110.(V.Thebault)

An arbitrary diameter D of the circumcircle of an equilateral triangle cutsthe sides BC, CA, AB in the points P , Q, R. Show that the Euler lines oftriangles AQR, BRP , CPQ determine a triangle T symmetrically equal toABC with the center of symmetry on D.


AMM3957.404.S421.(O.Dunkel)

Given ABC with A < B < C, show that there are precisely one, two, threestraight line segments which bisect both its perimeter and area according as

1 − sinAsinB

≥,=, or ≥ 2 tan2 A

2tan2 B

2

where we may replace B by C. If B = C, there are one, two, or three suchsegments according as

A ≤,= or ≥ A0

where sin A)

2 =√

2 − 1.

AMM3959.405.S4110. (N.A.Court)

The four pairs of reciprocal transversals a, a′; b, b′; c, c′; d, d′ are situ-ated, respectively, in the faces BCD, CDA, DAB, ABC of the tetrahedronABCD.

(1) If the lines a, b, c, d are coplanar, so also are the kines a′, b′, c′, d′.(See, Court, Modern Solid Geometry, p.121).

(2) If the lines a, b, c, d form a hyperbolic group, so also do the remainingfour lines.

AMM3960.405.S4110. (R.E.Gaines)

If a series of triangles be constructed so that the sides of each are equal tothe medians of the following one, then

(1) the area of each triangle is three - fourths of the area of the onefollowing;

(2) the alternative triangles are similar;(3) excluding the case of equilateral triangle, no two consecutive triangles

of the series are similar.


Each face angle of a given trihedral angle O − XY Z is π3 , and on the re-

spective edges the points A, B, C are located. Show that the Monge pointof the tetrahedron OABC describes a sphere as A, B, C vary on the edgesso that OA2 +OB2 +OC2 remains constant.



A circle (C) with a given radius rolls on a fixed circle (C ′). Find the form ofthe locus of the points of intersection of (C) with the polar of a fixed pointP with respect to (C). Consider the cases where (C ′) reduces to a point, ora straight line.


In an orthocentric tetrahedron the first sphere of twelve points is the locusof points the sum of whose powers, with respect to the spheres having asdiameters the edges (or bimedians), is zero. Generalize.


The sum of the powers of the vertices of a tetrahedron, with respect tothe Monge sphere of the circumscribed ellipsoid of Steiner, is equal to thenegative of half the sum of the squares of the edges.


For a given triangle ABC and a second triangle A′B′C ′ is formed whereAA′, BB,, CC ′ are segments of altitudes and AA′

BC = BB′CA = CC′

AB = k.(1) Show that the two triangles have the same centroid.(2) Examine the variation of the area of A′B′C ′.(3) For what value of k do the two triangles have the same angle of

Brocard ?(4) If k ± 1, show that the centers of squares constructed exteriorly, or

interiorly, on the sides of A′B′C ′ are the vertices of ABC.

AMM3968.40p574.S423. (F.Aryes)

Let the line through the verex Ai, (i = 1, 2, 3), and parallel to the oppositeside of the triangle A1A2A3 meet the circumcircle in the point Di. Showthat

1. the 2−lines [AMM3929.30p601] of the pairs of points Ai, Di inter-sect on the nine-point circle of A1A2A3 midway between Ai and theorthocenter of the given triangle;

2. the 2−lines of Di intersect in the symmetrics of A1, A2, A3 as to thenine-point center.


AMM3969.408.S423. (F.Aryes)

Let the line through the verex Ai, (i = 1, 2, 3), and parallel to the oppositeside of the triangle A1A2A3 meet the circumcircle in the point Di. Showthat

1. the orthocenter of D1D2D3 lies on the join of the circumcenter andisogonal conjugate point of the nine-point center of A1A2A3;

2. the join of the orthocenters of D1D2D3 and A1A2A3 is the image lineof the Steiner point of the latter triangle.

AMM3970.40p574.S434. (V.Thebault)

Let (H) ≡ A1A2A3A4A5A6 and (D) ≡ A1α1A2α2 · · ·A6α6 be a regularhexagaon and a regular dodecagon inscribed in a circle (O). Show that

(1) The Simson lines 1, 2 of any point M on (O) with respect to thetriangles A1A3A5, A2A4A6 are perpendicular and intersect at the midpointof MO.

(2) The consecutive sides of the pedal (H ′) of M with respect to (H) areparallel to 1, 2.

(3) The opposite sides of the pedal (D′) of M with respect to (D) areparallel to the bisectors of the angles between 1 and 2.

(4) Two sides of (D′), separated by a side, are perpendicular.(5) If we denote by S6, S12, Σ12 the areas of (H), D), (D′) then Σ12 =

S6 + 12S12.

(6) Extend (3) and (4) to pedal polygons of M with respect to a regularpolygon of 6k sides, k being any integer.

See also E3861, E3886.are perpendicualr and intersect at the midpoit of MO.

AMM3971.409. (J.R.Musselman)

If O be the circumcenter of the triangles A1A2A3 and Mi be the other pointsof intersection of the circle with the lines OAi, show that the trhee circlespassing through O, and on Mi with centers on AjAk respectively, meet atthe point of Feuerbach for the tangential triangle of A1A2A3.

AMM3972.409.S423. (N.A.Court)

With the traces of a plane on the edges of a tetrahedron as centers, spheresare drawn orthogonal to the circumsphere of the tetrahedron. Show that


the twelve points of intersction of the six spheres with respective edges forma desmic system.


The symmetrics of the Apollonian circles of a triangle with respect to thecorresponding midpoints of the sides are orthogonal to the circumcircle ofthe anticomplementary triangle. (N.A.COurt has given other properties ofthese circles in AMM, 1926, p.373.)


solid geometry.

AMM3975.4010.S424. (R.Goormaghtigh)

The orthopole of a straight line parallel to one of the axes of equilateralhyperbola as to the four triangles formed by any three of four points givenon that hyperbola are on a straight line.

AMM3976.4010.S424. (W.O.Pennell)

Through a point P in the plane of a central conic, a line CC ′ is drawn parallelto the diameter conjugate to the diameter located by a line joining P withthe center of the conic. Draw two lines through P intersecting the conic inA and B, and A′ and B′, respectively. Prove that AB′ and A′B (extendedif necessary) intersect CC ′ at points equidistant from P . Likewise, AA′ andBB′ intersect CC ′ at points equidistant from P .


Given an orthocentric tetrahedron and the spheres which are loci of pointssuch that the ratio of the squares of their distances to the extremities of oneof the edges of the tetrahedron is equal to the ratio of the sum of the squaresof the edges of the faces containing the opposite edge. Show that the sumof the powers of the respective extremities of the latter edge with respecetto one or the other of the two spheres is constant.

AMM3979.411.S424. (W.V.Parker)

equation of circle.


Pappus theorem: If M is any point on the circle that passes through A1,A2, A3, A4, and hij represents the distance from M to the side AiAj , then

h41h23 = h12h34.


Let Si, i = 1, 2, . . . , 6 be the spheres of similitude of the spheres with centersat the verteices of the tetradheron A1A2A3A4 such that the square of theradius of any one of the latter is equal to one - half of the sum of the squaresof the edgges of the opposite face.

(1) Examine the relative positions of the spheres Si.(2) Show that these six spheres are orthogonal to the circumspheres of

A1A2A3A4 and of its anitcomplementary tetrahedron.(3) The powers of the extremities of an edge of A1A2A3A4 with repsect

to the sphere Si whose center is on the opposite edge are independent of thelength of that last edge.

(4) The sphere S′i symmetric to the spheres Si with repsect to the mid-

point of the edges upon which they are cnetred intersect in two pointscollinear with the cirumcenter of A1A2A3A4.

AMM3982.411.S425.(V.Thebault)

The vertices of the tetrahedron ABCD are centers of spheres the squares ofwhose radii are equal respectively to one - third of the sum of the squares ofthe edges through the considered vertex. Show that the sphere orthogonalto the four spheres is concentric with the twelve-point sphere of ABCD.

See, for example, N.A.Court, Modern Pure Solid Geometry, p.250, forthe twelve - point sphere of a tetrahedron.


The vertices of the tetrahedron ABCD are centers of spheres the squares ofwhose radii are equal respectively to k times the sum of the squares of theedges of the face opposite to the vertex considered, and they are also centersof spheres the squares of whose radii are equal respectively to k times thesum of the squares of the edges through the considered vertex. Let ω1 andω2 be the centers of the spheres, radii R1 and R2, orthogonal respectivelyto the two sets of four spheres, G the centroid, and O the circumcenter ofABCD.


(1) Show that the points O, G, ω1, ω2 are collinear and determine theirrelative positions.

(2) Show that 1k (R2

1 −R22) remains constant when k varies.

AMM3984.412.S425.(R.Goormaghtigh)

The two points P,Q are symmetric as to the circumcenter of a triangle, theisogonal conjugates of P , Q are P ′, Q′, and M is the midpoint of P ′Q′.Prove that PQ · P ′Q′ = 4R ·HM .


The six points P , Q, R, P ′, Q′, R′ are taken respectively on the edges BC,CA, AB, DA, DB, DC of the tetrahedron ABCD; and the radical aplanesof the circumsphere of ABCD and the spheres AQRP ′, BRPQ′, CPQR′,DP ′Q′R′ cut the planes of the faces BCD, CDA, DAB, la, lb, lc, ld. Showthat these four straight lines are in the same plane if PP ′, QQ′, RR′ areconcurrent, and conversely.


AMM3988.413.S426. (N.A.Court)

The symmetrics of a given straight line (plane) with respect to the sides(faces) of a given triangle (tetrahedron) form a second triangle (tetrahedron)perspective to the first, and the center of perspectivity is equidistant fromthe sides (faces) of the second triangle (tetrahedron).

AMM3989.413.S426. (N.A.Court)

Three given spheres with non-collinear centeres are touched by a fourthsphere in the points P , Q, R and (p), (q), (r) are great circles, in parallelplanes, on the three given spheres. Show that the three cones P (p), Q(q)and R(r) have a circle in common.

AMM3990.413.S483. (Thebault)

Let A′, B′, C ′ be the centers of squares BCA′1A

′2, CAB

′1B

′2, ABC

′1C

′2 con-

structed interiorly on the sides of trianlge ABC with centroid G and theangle V of Brocard. If cot V = 7

4 , show that:(1) the centers A′′, B′′, C ′′ of the squares constructed interiorly on the

sides of A′B′C ′ lie on a straight line through G.


(2) The angle V ′ of Brocard of A′B′C ′ is such that cotV ′ = 2.(3) The straight lines joining A, B, C respectively to the midpoints of

A′1A

′2, B

′1B

′2, C

′1C

′2 are parallel.

(4) The distance of the circumcenter from the orthocenter of the orthictriangle is equal to one fourth of the perimeter of the last triangle.


Four straight lines li in a plane determine a complete quadrilateral (Q)forming four triangles with orthocenters Hi, i = 1, 2, 3, 4. Show that theorthopoles of the straight line containing these four points, with respectto the four triangles, of the parallels to li through Hi are the orthogonalprojections of the Miquel point of the sides of (Q).

See also E3839.


Show that the envelope of a variable sphere (S) which has its center on aquadric surface of revolution and which is orthogonal to a sphere (Σ) tangentto (Q) along a circle is composed of two spheres passing through (C).

AMM3993.414.S427. (N.A.Court)

A variable plane passing through a fixed point of the face ABC of thetetrahedron DABC meets the edges DA, DB, DC in the points P , Q,R.Show that the locus of the point U common to the three planes PBC,QCA, RAB is a cone of the second degree.


A sphere (S) is tangent to the faces of a tetrahedron ABCD at the pointsA′, B′, C ′, D′ and the straight lines AA′, BB′, CC ′, DD′ are concurrentin the point P . The cones ΓA, ΓB, ΓC , ΓD, with vertices at A, B, C,D circumscribe (S). The planes through P parallel to the planes B′C ′D′,C ′D′A′, D′A′B′, A′B′C ′ cut the respective cones in four circles which lie ona sphere concentric with (S).


solid geometry.


AM4006.418. (P.D.Thomas)

AMM4007.418. (J.W.Clawson)

4008.418. (V.Thebault)

AMM4011.419. (N.A¿Court)

The pairs of straight lines a, a′; b, b′; c, c′; d, d′ are isogonal conjugates forthe trihedral angles A, B, C, D of the tetrahedron ABCD. Prove that:

(1) If the lines A, b, c, d are concurrent, so also are the remaining fourlines. (See Court, Modern Solid Geometry, p.242).

(2) If the four lines a, b, c, d form a hyperbolic group. so also do theremaining four lines.


Determine the kind of tetrahedron for which(1) the circumcenter (or incenter) is one of the medians;(2) the straight line joining the circumcenter to the centroid is perpen-

dicular (or parallel) to one of the faces.

AMM4014.4110.S431. (Erdos)

Show that, if S1 and S2 are two squares contained in the unit squares sothat they have no point in common, the sum of their sides is less than unity.

It is very likely true that, if we have k2 +1 squares contained in the unitsquare so that no two of them have a point in common, the sum of theirsides is less than k.

AMM4015.4110.S431. (N.A.Court)

If the base of a variable tetrahedron is fixed and the opposite vertex varieson a fixed sphere, the volume of the tetrahedron is numerically equal to thepower of the variable vertex with respect to another fixed sphere.


The points D, E, F are taken on the sides BC, CA, AB of a triangle ABC,and the points P , Q, R are then taken on the straight lines AD, BE, CF sothat AP

AD = BQBE = CR

CF = k PDAD = QE

BE = RFCF = λ. that the area σ of triangle


PQR is given by

σ = k2(2 + ′) + (1 − 3k)′λ(2λ− 1) + (1 − λ)2′,

where and ′ denote the areas of ABC and DEF respectively.Deduce from this that in a complete quadrilateral the midpoints of the

the three diagonals are collinear.

AMM4018.421.S432. (N.A.Court)

IF four points taken in the four faces of a tetrahedron are collinear, thiertrilinear polars for the respective faces cannot be (a) coplanar, (b) hyper-bolic.

AMM4019.421.S432,438. (R.Robinson)

Given a triangle ABC, prove that the bisectors of the interor and exteriorangles at C, the side AB and its perpendicular bisector, and the perpendic-ulars to AC at A and to BC at B, are all tangent to a parabola. Locate itsfocus.

AMM4024.422.S433. (N.A.Court)

AMM4025.422.S432 (V.Thebault)

Let A′1,

′2, . . . , A′

n be the vertices of equilateral triangles constructed exter-nally (or internally) on the sides A1A2, A2A3, · · · A2nA1 of a plane polygonof 2n sides (P ) ≡ A1A2 · · ·A2n, and M1, M2, . . . , Mn be the midpoints of theprincipal diagaonals A1An+1, A2An+2, . . . , AnA2n of (P ). The midpointsM ′

1, M′2, . . .Mn of the principal diagonals A′

1A′n+1, A

′2A

′n+2, . . . , A′

nA′2n

of the polygon (P ′) ≡ A′1A

′2 · · ·A′

2n are the vertices of equilateral trianglesconstructed upon th sides of the polygon (p) ≡M1M2 · · ·Mn.

Generalize by replacing the equilateral triangles by similar isosceles tri-angles.

AMM4026.422.S435,471. (Thebault)

(1) Construct a triangle ABC knowing a, A, and given that the medianand the symmedian from A are perpendicular and parallel to two givendirections.

(2) Indicate the properties of this special triangle.


(3) Let B′ and C ′ be the orthogonal projections of B and C on a variablestraight line AP which buts BC in P . The locus of the harmonic conjugateof P with respect to B′ and C ′ is a right strophoid having the vertexA fora double point and tangent to the hisectors of angle A.

AMM4029.42p202.S436. (H.L.Dorwart)

Let d1, D2, d3, d4 be the distances in order from the sides of a square oflength k units to any interior point P . Then

1k(√d1d2 ±

√d3d4) and

1k(√d1d4 ±

√d2d3)

represent the sines and cosines of two angles θ1 and θ2, since the sum of thesquares of these expressions

1k2

(d1d2 + d3d4 + d1d4 + d2d3) =1k2

(d1 + d3)(d2 + d4) = 1.

Give a geometric interpretation for the angles θ1 and θ2.


On the sides AB, BC, CD, DA of a convex quadrangle ABCD equilateraltriangles with vertices A′, B′, C ′, D′ are constructed exteriorly (or interi-orly). Show that the diagonals A′C ′ and B′D′ of quadrangle A′B′C ′D′ areperpendicular (or equal) according as the diagonals AC and BD of ABCDare equal (or perpendicular), and conversely.


On the sides A1A2, A2A3, . . . , A6A1 of a convex hexagon having equalprincipal diagonals squares with centers A′

1, A′2 . . . , A′

6 are constructedexteriorly (or interiorly). Show that in the hexagon formed by these centersthe sum of the squares of two opposite sides and of the principal diagonalwhich does not end in vertices of the two sides considered is a constant.Generalize for a convex polygon of 2n sides whose principal diagonals areequal.


The point M is chosen arbitrarily on a bisector of angle A of ABC, andM ′ is the isogonal conjugate. Show that the two circles each through Mand M ′ and tangent to the side BC are tangent also to the circumcircle ofABC.


AMM4039.425.S438. (N.A.Court)

The circumcenter of a tetrahedron (T ) and any point M are isogonal conju-gae with respect to the tetrahedron formed by the centers of the four spherespassing through M and the circumcircles of the faces of (T ).

AMM4043.426.S445. (H.F.Sandham)

Prove that the angle in which the major auxiliary circle of a conic inscribedin a triangle cuts the nine-point circle, is equal to the angle which the foci ofthe conic subtend at teh inverse of one of them in the circumcircle. Completethis result and deduce that the minor auxiliary circle of the conic which hasthe Brocard points as foci, touches the nine-point circle, and the majorauxiliary circle cuts the latter in an angle which is the complement of threetimes the Brocard angle of the triangle.

The angle between the pedal circle of a point and the nine-point circleis the complement of the sum of the angles each line joining the point to avertex makes with an adjacent side, no side being taken twice.

See also AMM3857.

AMM4044.426.s439. (V.Thebault)

Determine the straight lines such that the circumsphere of the pedal tetra-hedron of each of its points with respect to any given tetrahedron ABCDpasses through a fixed point P .

Examine the case for which ABCD is orthocentric and P is the foot ofone of its altitudes.


In an orthocentric tetrahedron ABCD the straight lines joining th centroidwith the circumcenter of the triangles of faces cut he respective radicalplanes of the circumsphere and the spheres with the medians of ABCD asdiameters in four points of the same plane.

AMM4053.428.S4310. (E.P.Starke)

Show that all triangles inscribed in an ellipse and having their centroids atthe center of the ellipse have the same area, which is the greatest possiblearea for the an inscribed triangle.


Show that all triangles circumscribed about an ellipse and having theircentroids at the center of the ellipse have the same area, which is the leastpossible area for the circumscribed triangle.

AMM4056.429.S442. (J.R.Musselman)

Let the line of images of any point T on the circum[circle] of triangle A1A2A3

cut the side AjAk in the point A′i. The perpendiculars to the sides AjAk at

A′i form the triangle B1B2B3; show that the straight lines AiBi meet in T .

(Editorial note: The line of images of T is the directrix of a parabola withthe focus T and tangent to the sides of A1A2A3.)

AMM4057.429.S443. (J.R.Musselman)

Let B1, B2, B3 be the points symmetric to the vertices of A1A2A3 in itscircumcenter O, and let C1, C2, C3 be the reflections of Ai in the perpen-dicular bisector of the sides of A1A2A3. It is known that the circles OB1C1,OB2C2, OB3C3 meet at a point P . Show that P lies on the Euler line ofA1A2A3 and thatO is the midpoint of PD, where D is the inverse in thecircumcircle of the orthocenter H of A1A2A3.

Solution by Eves using inversion with respect to O and power −R2.


Let D, E, F be the points of contacts of the inscribed circle (I) with the sidesBC, CA, AB of triangle ABC, and A′, B′, C ′ the feet of its altitudes. Showthat the distances of the points of intersection of the pairs of straight linessuch as B′C ′, EF from the radical axis of (I) and the nine-point circle of atriangle ABC are inversely proportional to the distances of the Feuerbachpoint from the feet of the altitudes.


If a point P is the orthopole of the three sides of a triangle A1B1C1 withrespect to another triangle A2B2C2 inscribed in the same circle as the first,the product of its distances from the sides of the first triangle is equal tothe similar product for the second.


AMM4063.42?.S443;457. (H.S.M.Coxeter)

In projective geometry the porism of triangles inscribed in one conic adn self-polar for another is commonly proved by showing that if one such triangleexists, we can find another with one vertex at any given point on the firstconic. This statement is easily seen to be valid in complex geometry. Discussits possible failure in real geometry.

AMM4064.42?.S442.


Let Ia, Ib, Ic, Id denote the centers of the spheres escribed in the truncatedtrihedral angles for the corresponding vertices of the tetrahedron ABCD,and A1, B1, C1, D1 the points where the straight line AIa etc. meet thefaces BCD etc. Show that

V ′

V= − ABCD

(S −A)(S −B)(S − C)(S −D),

V1

3V= − ABCD

(B + C +D)(C +D +A)(D +A+B)(A+B + C),

where V , V ′, V1 denote the volumes of the tetrahedra ABCD, IaIbIcId,A1B1C1D1; A etc denote the areas of the faces BCD etc. and 2S = A +B + C +D.

AMM4070.432.S444. (P.Erdos)

Let ρ denote the length of the radius of the inscribed circle of the triangleABC, let r denote the circumradius and let m denote the length of thelongest altitude. Show that ρ+ r ≤ m. (Correction: the proposer intendedto exclude obtuse angled triangles).


On the sides of the given triangle ABC directly similar triangles A1B1C1 etcare constructed interiorly. Determine these latter triangles so that (GA1)2+(GB1)2 + (GC1)2 is a minimum, where G is the centroid of ABC.


AMM4075.433.S448. (N.A.Court)

If the radical center of four spheres coincides with the Monge point of thetetrahedron (T ) determined by their centers, the tetrahedron (S) formedby the four radical planes of the given spheresith their orthogonal sphereis orthongonal to the twin tetrahedra (T ′) of (T ) (i.e., the perpendicularsdropped from the vertices of (S) upon the corresponding faces of (T ′) areconcurrent).


Given the triangle A1B1C1 show how to construct the triangle B1B2B3 sothat the triangle BjBiAk will be equilateral and exterior to B1B2B3.


If four spheres, each passing through a corresponding vertex of a tegrahedronABCD, intersect in pairs on the corresponding edge, th four spheres areconcurrent in a point M (S.Roberts). Show that (1) the points A′, B′, C ′,D′ diametrically opposite to A, B, C, D on the corresponding spheres arein a plane (P ) passing through M (R Bouvaist). (2) The plane (P ) is aSimson plane of the tetrahedron A1B1C1D1 formed by the planes parallelto the planes of BCD, CDA, DAB, ABC and passing respectively throughA′, B′, C ′, D′.

AMM4081.43?.S449. (O.Dunkel)

Through the vertices of the triangle ABC parallels Aα, Bβ, Cγ of arbitrarydirection are drawn meeting the transversal in the points α, β, γ; andthrough the latter points straight lines are drawn parallel respectively toBC, CA, AB rotated through the angle θ, thus forming a triangle A1B1C1

similar to ABC. Prove that

1. As the direction of the parallels varies the verteices A1, B1, C1 describethe straight line concurrent in a point φ(θ).

2. For the particular set of parallels Aα, Bβ, Cγ which have the directionof rotated through the angle −θ, the triangle A1B1C1 reduces tothe point φ(θ).

3. The locus of φ(θ) is a unicursal cubic passing through the circularpoints at infinity, the point at infinity of the Newton line (ABC,),


the orthopole of with respect to ABC, and A0, B0, C0, the pointsof intersection of the sides of ABC with .

AMM4084.435.S449. (O.Dunkel)

On the sides AjAk of a given triangle A = A1A2A3 as bases, directly similartriangles BiAjAk are constructed interiorly giving the triangle B = B1B2B3.Show that, if B has the maximum area when the sense of rotation of itsvertices is opposite to that for A, the triangle BiAjAk must be isosceleswith cotα cot V = 3, where α is the base angleand V is the Brocard anglefor A. Determine the form of the triangle B giving the maximum.


Given an equilateral hyperbola (H) and a circle (O) pasing through thecenter ω of (H), show that the necessary and sufficient condition for theexistence of an infinite number of triangles inscribed in (H) and circum-scribing the circle is that the center O ofthe circle lies on (H). Consider theenvelope of the sides of these triangles.

AMM4086.436. (P.Erdos)

Let A1, A2, . . . , A2n+1 be the vertices of a regular polygon, and O anypoint in its interior. Show that at least one of the angles AiOAj satisfiesthe relation

π(1 − 12n + 1

) ≤ AiOAj < π.

AMM4087.436.S4410. (B.Dick and B.M.Stewart)


If three circles A(ρ), B(ρ), C(ρ) with the same radius ρ are described inthe triangle ABC, and then the circles with centers A, B, C orthogonalrespectively to C(ρ), A(ρ), B(ρ); these three circles have the same radicalcenter M1 whatever the value of ρ. The same is true of three circles withcenters A, B, C orthogonal respectively to B(ρ), C(ρ), A(ρ), the radicalcenter being M2. If O is the circumcenter of ABC, show also that

1. the triangles ABC and OM1M2 have the same centroid;

2. the straight line M1M2 is perpendicular to the straight line throughthe centroid andthe Lemoine point;


3. if M ′1 and M ′

2 are the symmetrics of M1 and M2 with respect to O,then M1M

′1 and M2M

′2 are parallel to the Euler line.

AMM4089.437.S4410. (J.H.Butchart)

The tangent to Kiepert’s hyperbola at a vertex of the triangle is the harmonicconjugate of the symmedian with respect to the altitude and median fromthat vertex. It meets the corresponding side of the medial triangle at a pointon the tangent to the hyperbola at the centroid.

See also AMM3883,3883.

AMM4090.437.S4410. (N.A.Court)

AMM4092.437S482. (Thebault)

For the triangle ABC, let (A1B1C1), (A2B2C2), . . . , (AnBnCn) be the cen-ters of squares constructed exteriorly (or interiorly) on the sides (BC,CA,AB),(B1C1, C1A1, A1B1), . . . , of the corresponding triangles.

1. Show that the center ω1 of the circle orthogonal to the circles withcenters A B,C, and radii B1C1, C1A1, A1B1 coincides with teh centersof the nine-point circle of ABC.

2. Findthe locus of centers ω2, ω3, . . . , ωn of the circels, orthogonal tothe circles iwth centers A, B, C, and with radii (B2C2, C2A2, A2B2),(B3C3, C3A3, A3B3). . . .


If P is any point of a curve and Q is the corresponding point of the pedalwith respect to the point O, then OQ makes the same angle with the pedalthat OP makes with the curve.

AMM4102.4310.S452. (H.Demir)

Let O and I be respectively the circumcenter and incenter of a given triangleABC. Let A0, B0, C0 be points taken respectively on BC, CA, AB so thatthe sums of the algebraic distances of each point to two other sides are equalto a given length . Prove synthetically that

(1) the points A0, B0, C0 are collinear;(2) the sum of distances to the sides of ABC of points on A0B0C0 is the

constant ;(3) the line A0B0C0 is perpendicular to the line OI.


AMM4106.441.S453. (N.A.Court)

If the Monge point of a tetrahedron lies on the circumsphere, show that (a)the line joining the circumcenter to the centroid of a face is equal to half thecorresponding median of the tetrahedron; (b) each median subtends a rightangle at the Monge point. Conversely.

AMM4109.44?.S453. (N.A.Court)

Prove that the sum of the n2 powers of n given points with respeect to then spheres having for diameters the n segments joining the given points to avariable point in space is constant.

AMM4112.443.S457. (N.A.Court)

If three rulers, chosen arbitrarily, of the same system ofa gien hyperboloidaretaken for the edges of a parallelepiped, the diagonals of the parallelepipedmeet in a fixed point.


From a point P on the circumcircle of a triangle lines are drawn inclinedat angles θ to the sides of the triangle and meeting them in three collinearpoints. Prove that as P varies, the line on the three points envelops a three- cusped hypocycloid. Prove that this hypocycloid is the locus of a point ona circle, radius 1

2R sin θ, which rolls inside another circle, radius three timesthat of the rolling circle, whose center X is equidistant from the circumcenterC and the orthocenter O, and is such that angle OXC = 2θ, R being thecircumradius.

See also AMM4181.

AMM4116.444.S456. (N.A.Court)

Given the tetrahedron (T1) = SA1B1C1, the tangent plane to its circum-sphere at the diametric opposite of S meets the edges AS1, SB1, SC1 inthe points A2, B2, C2. The tangent plane to the circumsphere of the tetra-hedron (T2) = TA2B2C2at the diametric opposite of S meets the edgesof (T2) through S in the points A3, B3, C3 thus forming the tetrahedron(T3) = SA3B3C3 etc. Find the locus of the incenters of these tetrahedr[a].


AMM4117.444.S456. (J.Rosenbaum)

A polygon A1A2 · · ·An may be transformed into a polygon B1B2 · · ·Bn bylocating the points Bi on the sides AiAi+1 so that the ratio of AiBi to BiAi+1

is equal to a constant r. Prove that if T1 and T2 are two such transformationsfor the ratios r1 and r2, then T1 T2 = T1 T1, and generalize.


The straight lines joining the vertices of a triangle to the points of contactof the inscribed circle with the respective opposite sides meet in a point P .Show that the six points of contact of circles tangent to the two sides andorthogonal to a given circle with center P are on a circle concentric with theinscribed circle.

AMM4127.446.S471(corrected). (Thebault)

The straight lines AG, BG, CG, DG drawn through the vertices and cen-troid G of the tetrahedron ABCD meet its circumsphere again in A′, B′,C ′, D′, and the planes perpendicular to the respective lines at these latterpoints determine the tetrahedron A′

1B′1C

′1D

′1. Show that (1) the two tetra-

hedra have the same centroid and are hyperbolic; (2) the non-focal axis ofthe quadric surface with G as a focus inscribed in A′

1B′1C

′1D1 is equal to the

diameter of the orthoptic sphere of the Monge sphere of the Steiner ellipsoidinscribed in ABCD.

AMM4128.447. (C.E.Springer)

Consider the tangent planes to a sphere at three points A, B, C of a curvelying on the sphere. Let R be the limiting point of intersection of the planesas B and C move independently along the curve andapproach coincidencewith A. Each curve on the sphee through A has its corresponding R point.Prove that the curves through A, the locus of whose R points is a certain linelying in the tangent plane to the sphere at A, are the loxodromes throughA.

AMM4129.447. (F.C.Gentry)

Prove that the vertices of the original tetrahedron and those of either ofthe other tetrahedra of a desmic system of tetrahedra are the centers of theeight spheres which touch the faces of the third member of the system.



Let ABCD be a skew quadrangle; planes perpendicular at A to AC, at Cto CB, at B to BD, at D to DA fom a tetrahedron A1C1B1D1 with thecentroid G1. Show that the powers of A with respect to the sphere (GC) onG1C as diameter, of C with respect to (G1B) of B with respect to (G1D),of D with respect to (G1A) are equal.

AMM4131.448.S4510. (H.Demir)

Let C11C

12C

13 be the inscribed triangle of a reference triangle A1A2A3, and

C21C

22C

23 be that of C1

1C12C

13 , and so on, obtaining a triangle Cn

1Cn2C

n3 after

n steps. Denoting the angles of the nth triangle by Cni , prove that

1. Cni −π

3Ai−π

3= (−1)n2−n.

2. the limit of the direction of Cn2C

n3 as n → ∞, is the direction of one

of the trisectrices of the angle (A2A3, C12C

13 ), and from that observe a

method of trisecting an angle by ruler and complass in infinitely manysteps.

AMM4141.4410.S463. (H.S.M.Coxeter)

Prove synthetically that the four lines of AMM4018.432. are in general fourtangents to a twisted cubic.

AMM4145.451. (H.Eves)

Find the positions of three non-overlapping circles in a triangle which havea maximum combined area.


Two circles varying in magnitude and position roll on two fixed circles. Findthe loci of their centers of similitude. If the straight line of their centers hasa constant direction, the midpoint of the segment of their centers describesa straight line.

AMM4149.452.S463. (N.A.Court)

The powers of the vertices of a tetrahedron (T ) with respect to the spheresdetermined by the centroid of (T ) and the circles of intersection of the


respectively opposite faces of (T ) with a sphere (L), of arbitrary radius,concentric with the circumsphere of (T ), are equal.


In a tetrahedron ABCD with orthocenter H, the perpendiculars at A to thefaces ACD, ABD, ABC meet respectively the planes HCD, HBD, HBCin (A1, A2, A3), and similarly for the points (B1, B2, B3), and so on. Showthat the planes A1A2A3, B1B2B3 . . . are perpendicular to the medians ofABCD.


Find the envelope of the axes of conics inscribed in a quadrilateral.


Show how to constuct four spheres passing through a given point and tangentrespectively to the planes of three faces of a given tetrahedron so that thepoints of contact are twelve points of the same sphere.

AMM4162.456. (H.S.M.Coxeter)

Prove, by the methods of real projective geometry, that if a projectivityP ∧ P ′ on a conic is not an involution, the envelope of PP ′ is a conic.

AMM4165.456;S488(corrected). (Thebault)

Let A and B be two fixed points of a given circle while C and D are twovariable points of the same circle such that the arc legnth CD remainsconstant. The orthogonal projection of D on AC is P , and on BD is Q.Show that, (1) the Simson lines of C and D for the triangles ABD andABC have fixed directions. (2) The centers of the circles tritangent toDPQ (inscribed and escribed) describe two Pascal Limacons.


Given the straight lines , ′ and a point F . A variable circle through F andthe intersection of ,′ cuts ,′ in A and A′ repsectively. The circle throughF and tangent to , ′ at A, A′ respectively meet again on a parabola tangentto and ′ and having F as focus.


AMM4169.457;462(corrected);S484. (Thebault)

The tangents at the vertices A, B, C of a given triangle to its Feuerbachhyperbola form a triangle whose conjugate circle is tangent to the ninepointcircle of ABC at its Feuerbach point.


The powers of the vertices A, B, C of a given triangle with respect tothe circle (ω1), (ω2), (ω3) are respectively (ka2, kb2, kc2), (kb2, kc2, ka2),(kc2, ka2, kb2), where a, b, c are the lengths of the sides of ABC.

1. Find the loci of the centers of ωi as k varies.

2. The circumcenter of ABC is the centroid of triangle ω1ω2ω3 whichremains similar to itself.

3. The straight line ω2ω3 is perpendicular to the join of the centroid andLemoine point of ABC.


The twelve point sphere of any tetrahedron is the locus of the points uchthat the sum of the squares of their distances to the vertices diminished bytheir powers with respect to the circumsphere, is equal to one third of thesum of the squares of the edges.


The poles of the medians of a triangle A1A2A3 as to its circumcircle are threepoints of a lines; those points where the external bisectors of the angles ofthe tangential triangle of A1A2A3 meet the opposite sides of this tangentialtriangle.

AMM4181.45?.S473. (P.D.Thomas)

Lines are drawn from a point P on the circumcircle of an equilateral triangleparallel to the three sides, thus determining six points, wo on each siderespectively.

(1) Prove that the six points thus determined lie by threeson two straightlines.

(2) If Q is the point of intersection of these two lines, find the locus ofQ as P moves on the circumcircle.


See also AMM4115.

AMM4182.45?.S474. (C.Cosnita)

Show that the envelope of the conics circumscribing a triangle and such thatthe angle between the asymptotes is constant, is a curve of the fourth degreebitangent to the line at infinity at the circular points and having the verticesof the triangle for double points.

See also AMM4225.474.


For a given tetrahedron the ratio of the distances of the Monge point andof the circumcenter to the common perpendicular to two opposite edges isequal to cos θ, where θ is the angle between the two edges.


For a given tetrahedron A1A2A3A4, if P is the centroid of its antipedaltetrahedron ABCD, its barycentric coordinates are inversely proportionalto the squares of its distances to A1,A2, A3, A4; and conversely.

AMM4193.463.S476. (H.Demir)

If on the sides of an arbitrary pentagon A1A2A3A4A5 the triangles BiAi+2Ai+3

are constructed such that BiAi+2//AiAi+1, and BiAi+3//AiAi+4, then thelines AiBi concur in a point C.


In each of the triangles formed by three of the vertices of a cyclic quadrilat-eral, we consider the projection of the orthocenter on the circumdiameterparallel to the Simson line of the fourth vertex of the quadrilateral withrespect o the triangle. The four projections form a quadrilateral inverselysimilar to the one given and are on a circle concentric to the circumcircle ofthat quadrilateral.


There are ten ways to divide six points on a circle into two groups of threesoas to form pairs of triangles with no common vertex. The midpoints of the


segments joining the orthopoles of a given straight line with respect to eachpair of triangle are ten collinear points.


Let A1, B1, C1 be vertices of equilateral triangles constructed exteriorly, orinteriorly, on the sides BC, CA, AB of a triangle, and let A2, B2, C2 be theintersections (BC1, CB1), (CA1, AC1), (AB1, BA1). Then, V and W beingthe first (or second) centers isogonal and isodynamic of ABC, show that

1VW

=1

A1A2+

1B1B2

+1

C1C2.


In a tetrahedron T = ABCD the perpendiculars from an arbitrary point Pto the planes of the faces meet again the peal spheres of P in A′, B′, C ′, D′.

1. The intersections of correspoinding faces of T and A′B′C ′D′ belong toa hyperboloid.

2. If P is on a sphere with center O and if the perpendiculars to the facesmeet it in A′′, B′′, C ′′, D′′, the orthologic center of T an A′′B′′C ′′D′′,other than P , is the focus of the inscribed paraboloid with the axisparallel to OP .


Given in a plane a triangle ABC and the fixed point P which is the centerof a variable circle (P ). Find the locus of the radical center of the circlespassing through A, B, C respectively, which have with (P ) the sides BC,CA, AB as radical axes. Consider the analogous problem for a tetrahedronand a sphere with fixed center, and show that the locus is a twisted cubicthrough the vertices and centroid of the tetrahedron.


A tetrahedron is given for which the difference of squaes of opposite edgesis the same for the three pairs.

1. The three medians are equal and the line joining the circumcenter tothe centroid is perpendicular to one of the faces.


2. One of the altitudes passes through the symmetric of the orthocenterof the face correspondign with respect to the circumcenter of the face.

3. The sum, or the difference, of the cosines of two opposite dihedrals ofthe tangential tetrahedron is the same for the three pairs.


The four products of the three sides of faces of a tetrahedron ABCD areproportional to the tangents ofthe half angles of the corresponding cones ofrevolution inscribed in the trihedral angles at the vertices.


If a right triangle has sides of integral lengths and the sum of the sidesforming the right angle is a square, then the sum of the cubes of these twosides is the sum of two squares. Can the hypotenuse be a square?

AMM4206.466;471(corrected). (Thebault)

Consider spheres with centers at the vertices of a tetrahedron ABCD andradii equal respectively to k times the sum of the squares of the three oppo-site edges. Show that the sum of the squares of the distances from the fourspheres to the center of the sphere orthogonal to the four spheres is equalto

[2(4k + 1)2]R2 − 2k(k + 1)Σ,

where R is the radius of the circumsphere and Σ the sum of the squares ofthe six edges. Consider particular cases.

AMM4208.466;471(corrected);S492. (Thebault)

Given an orthocentric tetrahedron. If two isogonal conjugate points are alsoconjugate with respect to the circumsphere, their pedal sphere is orthogonalto the sphere, belongint to the linear net determined by the circumscribedand conjugate spheres, and whose center is the complementary point of theorthocenter with respect to the tetrahedron.


If the parallels to the sides of triangle ABC drawn through a point P on thecircumcircle meet that circle again at A′, B′,C ′, the orthocenter H of ABC


and those α, β, γ of A′BC, B′CA, C ′AB are on a straight line perpendicularto the Simson line as to ABC; and the center of gravity of α, β, γ dividesinto the ratio 2 : 1 the distance from H to the circumdiameter parallel to.


Given a tetrahedron ABCD andan arbitrarily chosen point M :

1. The sum of the powers of the vertices, respectively, with respect tothree spheres on (MB,MC,MD), (MC,MD,MA), (MD,MA,MB),(MA,MB,MC), as diameters, is equal to the sum of the squares ofthe edges.

2. Construct the point M when the sums of the powers of the verticesA, B,C, D, relative to the corresponding set of three spheres, areproportional to given numbers α, β, γ, δ. Consider the case wherethese last four numbers are equal.

See AMM4064.


On the sides AB, CD of an arbitrary quadrangle ABCD isosceles trianglesare constructed with the same sense A′AB, C ′CD, with the base angle θ;and on thesides BC, DA the isosceles triangles B;BC, D′DA with the baseangle π

2 and in the same sense as the first. Prove that

1. The line A′C ′, B′D′ are perpendicular and that thelengths of the seg-ments are in the ratio tan θ.

2. The centroid of the vertices of the quadrangle is on th striagh linejoining the midpoints of A′C ′, B′D′, which it divides in the ratiocot3 θ.

3. Find the locus of the midpoints of the sides and the diagonals of thequadrangle A′B′C ′D′, and the envelope of all these lines.


In a tetrahedron T = ABCD, if a point L with normal coordinates (x, y, z, t)is such that its associates (−x, y, z, t), (x,−y, z, t), (x, y,−z, t), (x, y, z,−t)are on the circumsphere, it coincides with the point whose distances to


the planes of the faces BCD, CDA, DAB, ABC are proportional to theradii of the circyumcircles ofthese faces (second Lemoine point for T ), andconversely.


In an orthocentric tetrahedron ABCD with the altitudes AA′, BB′, CC ′,DD′, let H ′ be the inverse of the orthocenter H with respect to teh circum-sphere, which the lines H ′A, H ′B, H ′C, H ′D meet again in A1, B1, C1, D1.Show that the tetrahedra A1B1C1D1 and A′B′C ′D′ are similar and that thevolume of the first is 27 times that of the second.


In a tetrahedron the harmonic plane of the point L whose normal coordinatesare proportional to the radii of the circumcircles of the triangles of the faces(second Lemoine point), coincides with the polar plane with respect to thecircumsphere.


In a tetrahedron ABCD,(1) the right cones with vertices at the orthogonal projections of the

second Lemoine point L on the axes of the circumcircle of the faces andwith these circles as bases have the same base angle V (Brocard angle),

(2) The symmedians AL, BL, CL, DL met the circumsphere in thevertices of the tetrahedron A′B′C ′D′ having the same Brocard angle V andthe same Lemoine point as ABCD.

AMM4228.4610. (Thebault)

In a tetrahedron ABCD the straight lines joining each vertex to the pointsof intersection V and W of the six spheres of similitude of four given sphereswith centers A, B, C, D, meet the circumsphere in the vertices of two equaltetrahedrons A′B′C ′D′ and A′′B′′C ′′D′′.

AMM4231.471. (P.Nemenyi)

Show that any parabola y = axn, (a = 0, n > 0) has the following property:if through the vertex any ray is drawn, the ratio of the are of the segmentto


that of the largest inscribed triangle is independent of the direction of theray. Are there other curves with the same property?


Parallel lines, of arbitrary direction, through the vertices A′, B′, C ′, D′ ofa tetrahedron A′B′C ′D′ intersect in A1, B1, C1, D1 the faces BCD, CDA,DAB, ABC of a homethetic tetrahedron ABCD. If k is the homotheticratio, V the volume of ABCD, and V1 that ot A1B1C1D1, then

V1 = −k2(2k + 1)V.

See also AMM4359.

AMM4234.472.S498.(Thebault)

Show that if the Monge point of a tetrahedron ABCD lies in the plane ofthe face BCD, then the altitudes through vertex A, of the triangles ABC,ACD, ADB are coplanar, and conversely. Examine the cases where theMonge point lies on an edge and at a vertex of the face BCD. Show how toconstruct tetrahedra illustrating each case.


AXBZ is a jointed rhombus connected with a fixed point O by two equalrods OA, OB. OCZD is a jointed rhombus and Y C, Y D are equal arods.(Two Peaucellier cells, as it were “cross joined”). Prove that, as Y describesa circle, S described a conic.


The point M , situated in the interior of a tetrahedron ABCD, such thatthe volume of the tetrahedron having for vertices the points of intersectionsof the lines AM , BM , CM , DM with the opposite faces of ABCD be amaximum, coincides with the centroid of ABCD.

AMM4245.474. (J.H.Butchart)

The envelope of two families of lines, PQ, PQ′ making angles of ±30 re-spectively with the tangents to a deltoid at their points of contact P aretwo deltoids, larger than the given one in the ratio

√3 : 1. Show also that

PQ = PQ′, where Q, Q′ are the points of contact of PQ, PQ′ with the


respective envelopes, and that the angles between the cusp tangents of theenvelopes and the included cusp tangent of the given deltoid are ±10.

AMM4248.474;477(corrected);S4910. (V.Thebault)

Having given a tetrahedron ABCD, place a sphere (S) of a given radiusin such a manner that the volume of the polar tetrahedron of ABCD withrespect to (S) will be a relative minimum.


In an orthocentric tetrahedron the linesjoining the symmedian points (Lemoinepoints) of the faces to the midpoints of the corresponding altitudes, are con-current at a point such that the sum of the squares of its distances to theplaens of the faces isa minimum.

AMM4253.475.S4810. (G.T.Williams)

Given two tangent unit circles, C1 and C2, and their common external tan-gent T . A third circle, C3 is drawn tangent to C)1, C2, and T ; C4 is thendrawn tangent to C2, C2 and C3; and so on, each Cj+1 being tangent to C1,C2 and Cj. Find the total area of the aggregate of circles C1, C2, C3, . . . .


Given a sphere orthogonal to two circles lying in twodistinct planes.If thecenter of the sphere is conjugate, with respect to one of the circles, to thepoint in which the plane of that circle cuts the axis of the other circle,the same is true of the center of the sphere, if the roles of the circles aerinterchanged. (Note: A circle is orthogonal to a sphere if the plane of thecircle cuts the sphere along a greate circle orthogonal to the given circle.)

AMM4258.47?.S491. (H.F.Sandham)

Prove that the necessary and sufficient condition that four non-collinearpoints are such that each is the orthocenter of the other three, is

±34 · 42 · 23 ± 41 · 13 · 34 ± 12 · 24 · 41 ± 23 · 31 · 12 = 0,

where rs denotes the distance between the rth and the sth points, and threeof the signs differ from the fourth.



In a triangle ABC inscribed two triangles A1B1C1 and A2B2C2 whose sidesare parallel to the medians. Show that

1. the triangles have the same centroid and the same Brocard angle;

2. the triangles A1B1C1, A2B2C2 are inscribed in an ellipse concentricand homothetic to the inscribed Steiner ellipse, the ratio of homothetybeing 1√

3.


If two tetrahedra are homothetic with respect to their common centroid, thetwelve point sphere of one of these tetrahedra is tangent to the twelve pointsphere of the four tetrahedra which rthe planes of its faces cut off from thetrihedral angles of the other tetrahedron.


Given a tetrahedron ABCD and a sphere (S). If the polar planes of thevertices A, B, C,D with respect to (S) and the corresponding planes tangentto the circumsphere at A, B, C, D cut each other, respectively, on the facesBCD, CDA, DAB, ABC, the tetrahedron is orthocentric. Establish aconverse.

AMM4271.479.S496. (N.A.Court)

The external bisectors of the three faces angles of each trihedron of a giventetrahedron are coplanar. The four planes form a second tetrahedron. Showthat the lines joining corresponding verties of the two tetrahedra form, ingeneral, a hyperbolic group.

AMM4274.479.S495. (R.Bouvaist)

Let A, B, C, D be arbitary points on an equilateal hyperbola (H), and letA′, B′, C ′, D′ be the corresponding diametrically opposite points.

1. The isogonal conjugates of A′, B′, C ′,D′ with respect to the trianglesBCD, CDA, DAB, ABC respectively, coincide in the same point P .

2. The isogonal conjugates of A, B, C, D with respect to the trianglesB′C ′D′, C ′D′A′, D′A′B′, A′B′C ′, respectively, coincide in the samepoint P ′.


3. P and P ′ are diametrically opposite on (H).


Prove that a tetrahedron ABCD, whose vertices A, B, C are the inverse,with respect to a sphere with center D, of a right - angled triangle, and thetangential tetrahedron A1B1C1D1 are each inscribed in the other and thatthe midpoints of AD1, BC1, CB1, DA1 are coplanar.


The lines joining the orthocenter of a triangle to the points of intersectionof the medians with the nine-point circle passes trhough the vertices ofparabolas tangent to two sides of the triangle and having the third side forchord of contact.


In a tetrahedron ABCD, the incenter of which is I, the perpendicular atA to the faces ACD, ADB, ABC respectively cut the planes ICD, IDB,IBC in A1, A2, A3.

1. The perpendicular throughA to the planeA1A2A23 passes through thepoint of contact ofthe inscribed sphere with the opposite face BCD.

2. The analogous property is true for a triangle ABC.


Consider similar triangles ABC, the sides BC, CA, AB of which passthrough the fixed points A1, B1,C1.

1. The locus of the circumcenters O of these triangle is a circle.

2. The circumcirles (O) are orthogonal to a fixed circle.

3. When the points A1, B1,C1 are collinear, the envelope of the circles isa cardioid. What is this envelope otherwise?


AMM4309.487. (R.Goormaghtigh)

Let ABC be a triangle, D, E, F the contact points of one of the tritangentcircles with BC, CA, AB respectively, and let A′, B′, C ′ and D′, E′, F ′ bethe projections of a point M of that circle on BC, CA, AB and EF , FD,DE. Show that the lines joiniing the projections of M on A′E′ and A′F ′,B′F ′ and B′D′, C ′D′ and C ′E′ are concurrent.

AMM4313.488 (Thebault)

Two perpendicular chords AB, CD of a circle O intersect at apoint P insidethe circle. There are eight circles O1, O2, O3, O4, O′

1, O′2, O

′3, O

′4, tangent

to both chords and also to the circle O, the first four exteriorly, and the lastfour interiorly, and such that Oi and O′

i lie in opposite quadrants formed bythe given chords.

1. The sum of the squaers of the distances O1O′1, O2O

′2, O3O

′3, O4O

′4 is

independent of the position of P , and the products (O1O′1)(O3O

′3) and

(O2O′2)(O4O

′4) are equal.

2. The radical axes of the circles O1, O′1, O3, O′

3 taken in pairs and thoseof the circles O2, O′

2, O4, O′4 taken in pairs, intersect one another in

thirty six points of which twelve are on the circle O, four of thesecoinciding with the vertices of the square whose diagonals are parallelto AB and CD.


Let there be given a skew quadrilateral having the sum of one pair of oppositesides equal to the sum of the other pair. Then

1. There are infinitely many spheres tangnet to all four sides of thequadrilateral, the locus of the centers being a striaght line .

2. The points of contact of any one of the sphers lie on a plane perpen-dicular to .

3. The sides of the quadrilateral belong to a hyperboloid of revolutionwhich envelopes all the spheres and has as axis.



Let ABCD be a convex quadrangle circumscribed about a circle with centerO, and let A′, B′, C ′, D′ ne the points of tangency of the sides BC,CD,DA, AB. COnsiderthe circle (OA,B), (OB,D), tangent to OA at O andpassing, respectively, through the vertices B, D neighbouring A. Consideralso the analogous circles tangent to OB, OC, OD at O.

1. The pair of circles (OA,B) and (OD,C) (OB,C) and (OA,D), (OC,D)and (OB,A), (OD,A) and (OC,B) resepctively intersect at N on BC,P on CD, Q on DA, M on AB.

2. The quadrangle MNPQ is a parallelogram with center O having itssides parallel to the diagonals of ABCD and its diagonals parallel tothose of A′B′C ′D′.

3. The centers of the circle and of the equilateral hyperbola circum-scribign A′B′C ′D′, and the Miquel point of the complete quadrilateralformed by the sides of the quadrangle ABCD, are collinear.


Given a triangle whose altitudes are AA′, BB′,CC ′. Prove that the Eulerlines of the triangles AB′C ′, BC ′A′, CA′B′ are concurrent on the nine-pointcircle at a point P which is such that one of the distances PA′, PB′, PC ′

equals the sum of the other two.

AMM4334.492. (H.F.Sandham)

Prove that the feet of the six perpendiculars from the Bennett point on thesides of a complete quadrangle lie on a conic.


(1) If, in a triangle, oen of the angles is 120 or 60, two of the Feuerbachpoints are diametrically opposite on the nine-point circle, and conversely.

(2) If the triangle is scalene and if the circle through the feet of theinterior bisectors (or one interior and two exterior bisectors) passes throughone of the vertices, three of the Feuerbach points form an isosceles triangle,and conversely.



A necessary and sufficient condition for a tetrahedron to be isosceles is thateach of two bialtitudes of the tetrahedron divide the opposite edges propor-tionally.


If in a tetrahedron one draws lines through the vertices parallel to a givendirection and then locate the homothetics of the intersections of these lineswith the circumspheres with respect to the centroids of the correspondingfaces, in the ratio −1

2 , the four points so obtained lie in a plane perpendicularto and passing through the Monge point of the tetrahedron.

See also AMM4233.

AMM4364.498.(J.Rosenbaum)

On the sides AiAi+1 of an n−gon A1A2 · · ·An as bases, isosceles trianglesAiAi+1Bi are constructed, either all exteriorly or all interiorly, with thevertex angle Bi = 360

n . Prove(a) If A1A2 · · ·An is a projection of a regular n−gon, then B1B2 · · ·Bn

is regular.(b) The problem of locating the points Ai when the points Bi are given

is a porism.

AMM4370.4910. (H.F.Sandham)

A′, B′, C ′ are points on the opposite sides of a triangle ABC. The circlesthrough B′C ′A, C ′A′B, A′B′C intersect in M , the Miquel point, whoseisogonal conjugate is M ′. Prove that M , M ′ are corresponding points underthe direct circular transformation set up by A, A′; B, B′; C, C ′.



Elementary Problems, 1950 – 1959

E992.51?.S52?,531.(K.Tan)

Draw a straight line which will bisect both the area and the perimeter of agiven convex quadrilateral.

E1014.52?.S531.(V.Thebault)

In ABC, if the ratio a2+b2+c2

4 is an integer > 1, then the sides a, b, ccannot all be integral.

E1017.52?.S532.(H.Furstenberg)

On th common secant AB of two intersecting circles, O and O′, are chosenany two points, C and D, outside of either circle. The tangents CQ and CSare drawn to O′ and O respectively, on one side of AB and the tangents DRand DT are drawn to O′ and O on the other side of AB. Prove that QRand ST intersect on AB.


Find all triangles for which the three sides and one of the altitudes are, insome order, measured by four integers in arithmetic progression.

Answer. Only the multiples of 13,14,15, with height 12 (on the side 14).See also E695.46?.

E1023.52?.S532.(D.J.Newman and H.S.Shapiro)

(1) Given any set of points in the plane, not all coincident with the origin,show that there exists a point on th unit circle such that the product of thedistances from it to these points is greater than 1.

(2) Given any set of points on the unit circle, not the vertices of aregular polygon, show that there exists a point on the unit circle product ofthe distances from it to these points is greater than 2.


E1027.52?.S533.(R.Clark and R.Oeder)

Find the total area enclosed by the set of circles formed as follows: Constructthe inscribed circle of a triangle ABC and draw the tangents to this circlewhich is parallel to a selected side of the triangle; construct the inscribedcircle of the new triangle so cut off and draw the tangent to this circle whichis parallel to the selected side of ABC; repeat the process indefinitely.Also show that the set of maximum area is obtaiend if the lines are drawnparallel to the shortest side of triangle ABC.

E1028.52?.S533.(V.Hoggatt)

Do there exist Pythagorean triangles whose sides are Fibonacci numbers ?Solution. Trivially no.

Remarks by W.F.Cheney: It is quite likely that (3,4,5) and (5,12,13) arethe only Pythagorean triangle two of whose sides are Fibonacci numbers.

E1030.52.S74,1110.7510.()

Consider the polygon formed by the internal trisectors of the angles of agiven n−gon, intersecting in neightboring pairs.

(a) Prove that a necessary and sufficient condition that the trisectorpolygon be regular is that the parent n−gon be regular when n ≥ 4.

(b) Prove that the area ratio between the parent and trisector polygonsis always irrational.

Counterexamples are given.


Show that if the circle passing through the feet of the symmedians of a non-isosceles triangle of sides a, b, c is tangent to one side, then the quantitiesa2 + b2, c2 + a2, a2 + b2, arranged in some order, are consecutive terms of ageometric progression.

E1039.52?.S535.(I.W.Burr)

Minimize the product of two perpendicular central chords of a given ellipse.

E1043.52?.S536.(O.J.Ramler)

Prove that the sum of the ratios in which a point within a triangle dividesthe cevians of this point is never less than 6 and that the product of the


ratios is never less than 8.

E1044.52?.S536.(J.E.Wilkins)

Findm−1∏r=1

(2 sin

πr

m

)r

.

E1052.532.S538.(H.H.Berry)

Let AOB be a fixed diameter of a given circle (O), and let P be any pointon the circle. Denote by Q the foot of the perpendicular from P on AB andby R the foot of the perpendicular from O to PA. Let PQ and RO intersectin N , and let QR and PO intersect NA in L and M , respectively. Find theloci of points L, M , and N as P moves along the given circle.


The centers of the four circles pasing through triples of vertices of a quadri-lateral ABCD inscribed in (circumscribed about) a circle (O) are the ver-tices of a quadrilateral A′B′C ′D′ inscribed in (circumscribed about) a circle(O′).

E1057.532.(Klamkin)

Find the sum of the first n terms of the series

sec θ +12

sec θ sec 2θ +14

sec θ sec 2θ sec 4θ + · · ·

E1059.533.(C.Y.Wang)

Let a circle and an inscribed closed polygon of n sides be given. Show thathte product of the distances of a point on the circumference of the circlefrom the sides of the polygon is equal to the product of the distances ofthe same point from the sides of the tangential polygon (i.e., the polygonformed by the tangents to the circle at the vertices) of the given polygon.

E1065.534.S5310.(C.S.Ogilvy)

Find the largest plane section of a given solid right circular cylinder.


E1066.535.S541. (A.Zirakzadeh)

Inscribe a trapezoid in a given quadrilateral such that the bases of the trape-zoid will be parallel to one of the diagonals of the quadrilateral and the othertwo sides will pass, respectively, through two given points.

E1068.535.S541. (W.O.Pennell)

Given a triangle with sides a, b, c and s2 = 2ab, where s is the semiperimeter.Show that

(1) s < 2a, s < 2b;(2) a > c, b > c.

E1069.535.S541. (R.Buehler, A.Gregory, and J.R.Wilson)

How un-isosceles can a triangle be ?

E1073.536,741(?).(G.W.Walker)

A polygonal spiral A1A2A3 · · · of unit segments winds counterclockwise andis construcrted in the following manner: Point A1 is at the origin, point A2

is at (1, 0), An−1AnAn+1 = 2πn for all n ≥ 2. Is there a point lying within

the interior of each An−1AnAn+1? If so, what are its coordinates?

E1080.537.S543. (V.Thebault)

Let I be the incenter, N thenine-point center, and D the midpoint of sideBC of ABC. Show that one of the common tangents to the circles I(N)and D(N) is parallel to BC.

E1085.538.(J.Langr)

The perpendicular bisectors of the sides of a quadrilateral Q form a quadri-lateral Q1, and the perpendicular bisectors of the sides of Q1 form a quadri-lateral Q2. Show that Q2 is similar to Q and find the ratio of similitude.

E1090.539.S545. (B.M.Stewart)

From one vertex of a triangle lines are to be drawn dividing the triangle intoa set S of n triangles having equal inscribed circles.

(1) Show that in general the set S may be constructed by ruler andcompass if and only if n = 2s.


(2) Show that the n−k+1 triangles formed by taking sets of k adjacenttriangles of the set S have equal inscribed circles (k = 2, 3, . . . , n− 1).

(3) Find a neat construction when n = 2.

E1092.5310.S546. (N.A.Court)

The homothetic center of the orthic and tangential triangles of a given tri-angle T is the pole of the orthic aaxis of T with respect to the circumcircleof T .

E1097.541.S547. (Bankoff)

Arcs AB and CD are quadrants of circles tangent externally at their mid-points, E, and such that AC and BD when extended meet perpendicularlyin F . A circle is inscribed in the mixtilinear triangle EDB, touching ED inM . G is the projection of M upon EF . Show that triangle MGF is a 3:4:5right triangle.

E1103.542.S548. (P.Monsky)

Find the locus of the vertex of a tri-rectangular trihedral angle which movesso that its edges intersect a given circle.

E1106.543.S549. (C.I.Lubin)

Two non-parallel, non-coincident lines which cut the circle |z| = r in thepoints a, b, and c, d respectively, where a, b, c, d are complex numbers notnecessarily all different, intersect in point z given by

z =a−1 + b−1 − c−1 − d−1

a−1b−1 − c−1d−1.

E1107.543.S549. (V.Thebault)

On the edges AB, AC, AD of a tetrahedron ABCD are marked points M ,N , P such that AB = nAM , AC = (n + 1)AN , AD = (n + 2)AP . Showthat the plane MNP contains a fixed line as n varies.

E1112=E1119.544.S5410. (L.C.Graue)

Consider two families of circles, one tangent at the origin to the x−axis andthe other tangent at the point (1, 1) to a line of slope m. Find the locus ofthe poitns of tangency of the two families.


E1115.544.S5410. (R.M.Gordon)

(1) Let Q1Q2Q3Q4 be a (not necessarily convex) plane quadrilateral. Onits sides construct similar isosceles triangles QvPvQv+1, with Q5 = Q1,having arbitrary base angle θ. The angles Qv+1QvPv(= θ) are orientedalternately clockwise and counterclockwise from the adjacent sides, Qv+1Qv

of the quadrilateral. Show that P1, P2, P3, P4, the vertices of the isoscelestriangle, are vertices of a parallelogram.

(2) Let P1P2P3P4 be a plane quadrilateral. On its vertices constructsimilar isosceles triangles QvPvQv+1, with Q5 = Q1, having vertex anglesQvPvQv+1 and base angle θ. The angles Qv+1QvPv = θ are oriented al-ternately clockwise and counterclockwise from the adjacent triangles, Showthat, for arbitrary θ, the bases of the isosceles triangles are the sides ofinfinitely many quadrilaterals Q1Q2Q3Q4, provided that P1P2P3P4 is a par-allelogram, and that if P1P2P3P4 is not a parallelogram then there exists aunique quadrilateral Q1Q2Q3Q4.

E1117.545.S551. (V.Thebault)

Construct a right triangle in which the legs and the altitude on the hy-potenuse can be taken as the sides of another right triangle. Solution. Re-quire 2a2 + b2 =

√5b2.

E1130.547.S553. (Thebault)

Let the perpendicular bisector of the median BB′ of triangle ABC and thetangent at B to the circumcircle of triangle ABC cut the line AC in pointsM and N respectively. Show that the triangle ABC is isosceles with vertexat A if and only if AM

AN = 34 .

E1132.548.S554. (Bankoff)

A common external tangent of two circles, tangent externally at C, cuts theirsmallest circumscribed circle in P and Q. The common internal tangent atC intersects the minor arc PQ in E, and the major arc QP in F . PCextended meets the outer circumference in K. Show that arc QK = arcKF .

E1138.549.S555. (J.P.Ballantine)

For any triangle prove that


(1) if B = 2A, then b2 = a2 + ac,(2) if B = 3A, then b3 − ab2 − a2b− ac2 + a3 = 0.

Editor’s Remark: Joyce Friedman’s method may be successively used toobtain relations for triangles having B = 4A, 5A, . . . . This raises theproblem of finding a general formula, involving the sides of the triangle,equivalent to B = nA, where n is a positive integer. In Problem E620.451,Wayne pointed out that the identity

sin(n+ 1)A = r sinnA− sin(n− 1)A,

where r = 2cosA, together with the law of sines, yields the equality

b

c=

1r−

1r− · · · 1

r

(to n component, where B = nA, and r = b2+c2−a2

bc .

E1139.549.S555. (N.A.Court)

Three collinear points P , Q, R are marked on the sides BC, CA, AB ofa triangle ABC. Starting with an arbitrary point X of the line BC, thefollowing points are constructed successively:

Y = (XR,CA), Z = (Y P,AB), X ′ = (ZQ,BC);Y ′ = (X ′R,CA), Z ′ = (Y ′P,AB), X ′′ = (Z ′Q,BC).

Show that points X and X ′′ coincide.

E1141.5410.S556. (Bankoff)

Find the radius of the circle inscribed in the mixtilinear triangle formedby the two legs of a given right triangle ABC and the semicircumferencedescribed externally upon the hypotenuse AB.

E1142.5410.S556. (Klamkin)

Find the semi-vertical angle of a right circular cone if three generating linesmake angles of 2α, 2β, 2γ with each other.

E1146.551.S557. (P.B.Johnson)

Show that any rectangle whose edges and diagonal are measured in integerscan be made the base of a rectangular parallelopiped whose three edges andmain diagonal are measured in integers.


E1147.551.S557. (E.P.Starke)

If cosα is rational (0 < α < π), prove that there are infinitely many triangleswith integer sides having α as one angle. In particular, given cosα = r

s , finda three- parameter solution for the ;sides a, b, c.

E1148.551.S557. (Thebault)

Let a, b, c be arbitrary points on th sides BC, CA, AB of triangle ABC, andlet A′, B′, C ′ be the reflections of A, B, C in the midpoints of the segmetnsbc, ca, ab. Show that the triangles abc and A′B′C ′ have equal areas.

E1153.552.S558. (Thebault)

For any angle θ, show that arbitrarily small constructible angles φ eixst suchthat θ − φ can be trisected.

E1154.552.S558. (Thebault)

The distance from the midpoint of side AB of a regular convex heptagonABCDEFG, inscribed in a circle, to the midpoint of the radius perpen-dicular to BC and cutting this side, is equal to half the side of a squareinscribed in the circle.

Proposer’s remark: LetO be the center of regular heptagon ABCDEFG,W the midpoint of OF , M the point diametrically opposite F , U the mid-point of AB, V the midpoint of OM , and J the point on UB produced suchthat UJ = UM . Then

(1) UW is equal to the diagonal of the square constructed on an apothemof the heptagon as a side.

(2) OJ is equal to the diagonal of the square constructed on half the sideof the inscribed equilateral triangle.

(3) UV is tangent to the circle through U , V , W .

E1160.553.S559. (H.Demir)

Prove that in a complete quadrilateral the isometric line of any side withrespect to the triangle formed by the other three is parallel to the Newtonline of the quadrilateral.


E1162.554.S5510. (Thebault)

Find two noncongruent similar triangles having two sides of one equal totwo sides of the other.

Solution. Let a < b be two numbers such that ba <

12(√

5 + 1), so thatthere is a triangle with sides a2, ab, and b2. Scaling this by factors 1

a and 1b ,

we obtain two noncongruent, similar triangles of sides a, b, b2

a , and a2

b , a, b.

E1165.554.S5510. (A.Sobczyk)

A vertex V of a closed polygon C having an odd (even) number of sidesis regular in case a triangle formed by extending the sides incident on Vand having for base a line segment containing th opposite side (vertex) to Vcircumscribes C. SHow that every convex C has at least one regular vertex.

E1166.555.S561. (Bankoff)

Let DE be a variable chord perpendicular to diameter AB of a given circle(O). The maximum circle (ω0) inscribed in the smaller segment, DEBtouches chord DE in C. The circle (ω1) is tangent to (ω0), (O), and DC,and another circle (ω2) is tangent to (ω1), (O), and DC. Find the ratio BC

CAfor which the radius of circle (omega2) is a maximum.

E1168.555.S561. (R.R.Phelps)

In an analogy with perfect numbers, let us define a perfect triangle as onewhose integer valued sides added up to twice its area. An example is the(3,4,5) triangle. Find all perfect triangles. Solution. This is the same assolving s = (s − a)(s − b)(s − c). Note that s = (s− a) + (s − b) + (s − c).We are finding three positive integers whose sum and product are the same.The only solution is clearly 1, 2, 3. This gives the (3,4,5) triangle as the onlyanswer.

Note: If xyz = x+y+z, then 1xy + 1

yz + 1zx = 1. At least one the numbers

xy, yz, and zx should be ≤ 3. From this, two of the numbers must be 1,2;or 1,3.

The only triangles with area equal to perimeter are (5, 12, 13), (6, 8, 10),(6, 25, 29), (7, 15, 20, and (9, 10, 17).


E1170.555.S561. (V.Linis)

Show that there exists a centrally symmetric hexagaon inscribed in anyclosed convex curve such that the ratio of the respective areas is at least 2

3 .

E1173.556.S562. (R.A.Laird)

The sum of the lengths of five equal contiguous chords inscribed in a givencircular arc is 5280 feet. The length of the long chord of the given arc is5208 feet. Find, to the nearest inch, the length of the given arc.

E1175.556.S562. (G.A.Yanosik)

(1) Three mutually tangent spheres, with radii r1 < r2 < r3, rest upon ahorizontal plane. Find the radius R of the largest sphere which will slipthrough the space between the three given spheres.

(2) Given four mutually tangent spheres of slightly different radii r1 <r2 < r3 < r4, find the radius R of the largest fifth sphere which will fit inthe space which is more or less bounded by the four given spheres.

E1177.557.S563. (W.R.Utz)

Dessribe three types of plane loci the product of whose distances from apoint and a line is constant.

E1178.557.S563. (A.J.Goldman)

Prove that there exists a positive constant c with the following property: ifT is any triangle whose area exceeds c, then the proudct of the lengths ofthe sides of T is greater than the area of T . What is the best possible valueof c? Solution. (C.Foreman) The product P of the lengths of the sides of atriangle is greater than the area if and only if R > 1

4 . The circumradius isminimal and equal 1

4 for a triangle of prescribed area when the triangle isan equilateral triangle whose area is 3

64

√3.

E1189.559.S565. (N.A.Court)

If two pairs of spheres with noncoplanar centers are such that each sphereof one pair is orthogonal to the two spheres of the other pair, then thetetrahedron formed by teh centers of similitude of the two pairs of spheresis orthocentric.


E1192.5510.S566. (E.Karst)

Let AB, BC be two adjacent sides of a regular nonagon inscribed in a circleof center O. Let M be the midpoint of AB and N the midpoint of the radiusperpendicular to BC. Show that angle OMN = 30.

E1193.5510.S566. (L.C.Barrett and H.Knothe)

(1) An ellipse has one focus at the center of two concentric circles, is tangentinternally to the larger circle, and has at least one point in common withthe smaller circle. Find the length of the maximum major axis of all suchellipses.

(2) Let the inner of two concentric circles represent a homogeneous spher-ical mass and the outer a presecribed orbit. Determine the range of specificenergy values a particle may be givenif it is to traverse a plane ellipticalpath from the surface of the sphere and just reach, but not cross, the circu-lar orbit.

E1194.5510.S566. (Thebault)

Construct a triangle ABC given A, ma+ b, na+ c, where m and n are givenpositive integers.

E1197.561.S567. (H.Demir)

Let ABC be a right triangle and CH the altitude on the hypotenuse AB.Show that the sum of the radii of the inscribed circles of triangles ABC,HCA, HCB is equal to CH. Remark. Bankoff listed 24 properties associ-ated with this configuration.

E1201.562.S568. (C.S.Ogilvy)

What is the area of the maximum cross section of the unit cube?

E1202.562.S568. (Thebault)

Let O be an arbitrary point on an arbitrary line λ passing through thecentroid G of a tetrahedron ABCD. If λ cuts the plane BCD, CDA, DAB,ABC in A′, B′, C ′, D′, show that

A′OA′G

+B′OB′G

+C ′OC ′G

+D′OD′G

= 4.


E1207.563.S569. (R.D.Gordon)

Given two mutually perpendicular line s l1 and l2 in a plane, and a point Qin the plane located equally distant from l1 and l2. Determine the locus ofa point P in the plane if its distance from Q equals the sum of its distancesfrom l1 and l2.

E1209.653.S569. (H.Demir)

Let ABC be any triangle and (I) its incircle. Let (I) touch BC, CF at E′,F ′ respectively. Show that the anharmonic ratio D(E,F,E′, F ′) is the samefor all triangles ABC.

E1210.563.S569. (M.Goldberg)

Given two equilateral triangles of edges a and b.Show how to dissect themby straight cuts into a total of no more than 6 pieces which can be assembledinto another equilateral triangle. When the ratio of the larger to the smallersatisfies a

b ≥ √3, then 5 pieces suffice. When a

b = 43 , 4 pieces suffice.

E1214.564.S5610. (P.Payette)

Find the envelope of the family of ellipses of constant major axis having onefocus at a given point and the other focus on a given straight line.

E1215.564.S5610. (J.P.Ballantine)

State a necessary and sufficient condition for an ordered set of n line seg-ments to be the consecutive sides of an n−gon possessing an inscribed circle.

E1216.565.S571. (N.A.Court)

The area of the triangle formed by the midpoints of three (not necessarilyconcurrent) cevians drawn through the three vertices of a given triangle isequal to one fourth of the area of the triangle determined by the feet of thecevians.

E1222.566.S572. (Thebault)

If we designate by C1, C2, C3 the sides of the regular convex heptagon andof the two regular star heptagons inscribed in a circle of radius R, then

C21 + C2

2 + C23 = 7R2.


Remark. Bankoff supplied the following items of interest relative to thefigure of the problem. In the triangle ABC whose sides are a = C1, b = C2,and c = C3, we have

1. bc = a(b+ c), ac = b(c− a), ab = c(b− a).

2. cosA cosB cosC = −18 , sinA sinB sinC =

√7

8 .

3. sin2A = 3a−c4a , sin2B = 3b−a

4b , sin2C = 3c+b4c .

4. cos2A+ cos2B + cosC = 54 , sin2A+ sin2B + sinC = 7

4 .

5. cotV = cotA+ cotB + cotC =√

7, where V is the Brocard angle.

6. cos2B cosC + cosC cosA + cosA cos2B = 38 .

7. ha = hb + hc.

8. h2a + h2

b + h2c = 1

2(a2 + b2 + c2).

9. b2

a2 + a2

c2+ c2

b2= 5.

10. OH = OIa = R√

2, ra = R2 , IaH = R.

E1228.567.S573. (V.Linis)

Let n(P ) be the number of distinct lines through a point P dividing the areaof a given triangle into two equal parts. Show that the locus of all points Pwith n(P ) ≥ 2 is a region the ratio of whose area to the area of the giventriangle is an absolute constant.

E1229.567.S573. (M.P.Drazin)

Given any point O in the plane of a triangle ABC, let the sides A, b,c subtend angles A′, B′, C ′ at O, and let the distances farom O to thevertices of the triangle be a′, b′, c′. Show that the triangle with sides aa′,b′, cc′ has angles A′ − A, B′ − B, C ′ − C, and find the sextic polynomialrelation connecting a, b, c, a′, b′, c′.



Arbitrary parallel lines drawn through the vertices A, B, C of a triangle in-tersect the circumcircle in A′, B′, C ′. Show that A′′, B′′, C ′′, the symmetricsof these points with respect to the midpoints of BC, CA, AB, respectively,lie on a line perpendicular to the parallel lines and passing through a fixedpoint of the triangle.

E1233.568. (J.Andrushkiw)

Denote by the sides and inradius of a triangle a0, b0, c0, and r0. The pointsof contact form a new triangle whose sides and inradius are a1, b1, c1, andr1. Repeating the process one obtains the sequence are an, bn, cn, and rn.Show that

limn→∞

rnan

= limn→∞

rnbn

= limn→∞

rncn

=√

36.

E1239.569. (J.Langr)

Let Q′ = A′B′C ′D′ be the quadrangle formed by the orthocenters A′, B′,C ′, D′ of triangles BCD, CDA, DAB, ABC of a given convex quadrangleQ = ABCD. Show that

1. the vertices of Q and Q′ lie on a common equilateral hyperbola,

2. Q and Q′ have equal areas.

E1240.569. (H.Lindgren)

Find six-piece dissections of a regular dodecagon into a square and a Greekcross.


Show that the circle orthogonal to the circles inscribed in the squares ofcenters A′, B′, C ′ constructed exteriorly (or interiorly) on the sides of atriangle ABC is concentric with the nine-point circle of triangle A′B′C ′.


Determine the relation between the radius of the base and the altitude ofa right circular cone in which a trihedral angle can be inscribed whose faceangles are all equal to a given angle 2α.


Show that if two trihedral angles whose face angles are all equal to 2α and2α′ respectively, are inscribed in two right circular cones having a commonbase, then a necessary and sufficient condition for th radius of the commonbase to be a mean proportional between the altitudes of the cones is that

sin2α+ sin2 α′ =34.

E1250.571. (N.A¿Court)

Through a point G two secants GAD, GBC are drawn meeting a givencircle (H) in the points A, D; B, C. Show that the points E = (AB,CD),F = (AC,BD) are the centers of similitude of the two circles orthogonal to(H) and having for centers the harmonic conjugates of G for the pairs ofpoints A, D; B, C, respectively.

E1254.572. (R.Robinson)

Prove that if two conics intersect in four distinct points, these points areconcyclic if and only if the axes of the two conics are parallel or perpendic-ular.

E1256.573. (W.B.Anderasen)

Discuss the error involved in the following approximate trisection of a circu-lar arc AB. On chord AB locate C such that BC = 1

3BA and D such thatCD = 7

6AB. With D as center and DC as radius describe an arc to cut arcAB in the approximate trisection point E.

E1257.573. (N.A.Court)

(1) The medial triangle of each of the four triangles formed by the sidesof a complete quadrilateral (q) taken three at a time is homological to thediagonal trilateral of (q).

(2) The four axes of the four homologies coincide.

E1272.57?.S58p123,607. (Thebault)

If A, B, C are the angles of a triangle, show that

(sinA

2+ sin

B

2+ sin

C

2)2 ≤ cos2 A

2+ cos2 B

2+ cos2 C

2.


Bankoff observed that the earlier solution was incorrect. The statementthat “the minimum value for cos2 A

2 + cos2 B2 + cos2 C

2 for A2 + B

2 + C2 ≤ π

2 isattained when A

2 = B2 = C

2 ” was incorrect. For a counterexample, let A =B = 10, C = 160. Here,

∑cos2 A

2 = 2.0149, whereas with A = B = C,this sum is 2.25.

E1350.591.S597. (N.A.Court)

(a) The tangents to the ninepoint circle of a triangle T at the midpoints ofthe sides of T form a triangle homothetic to the orthic triangle of T . (b)The homothetic center of the two triangles is a point on the Euler line of T .

The vertices of the triangle bounded by the tangents are

[ b2 + c2 c2 − a2 −(a2 − b2)−(b2 − c2) c2 + a2 a2 − b2

b2 − c2 −(c2 − a2) a2 + b2

].

The center of homothety is the point

b2 + c2

b2 + c2 − a2:

c2 + a2

c2 + a2 − b2:

a2 + b2

a2 + b2 − c2.

E1366.59?.S601. (V.E.Hoggatt)

Show that if a, b, c form a triangle, then√a,

√b,

√c form a triangle.

More generally, f(a), f(b), f(c) form a triangle for any nonnegative,nondecreasing, subadditive function f(x) defined for x ≥ 0.

E1375.59?.S602. (L.D.Goldstone)

Construct a triangle given A, ma and ta.

E1376.59?.S603. (V.F.Ivanoff)

Show that if A is the area of a quadrilateral having sides a, b, c, d, anddiagonals e, f , then

16A2 = 4e2f2 − (a2 − b2 + c2 − d2)2.

See also AMM 46 (1939) pp.345 – 347.


E1384.59?.S604. (J.H.Butchart)

Construct a circle through two given points, separated by a given circle,which shall cut the given circle at the smallest possible angle.

E1394.59?.S606. (V.Thebault)

Given a trihedral angle and a point within it, construct the plane throughthe point which intecepts on the trihedral angle the tetrahedron of minimumvolume.

E1397.601.S607. (Bankoff)

Show that∑AI ≤∑

AH.

E1398.601.S607. (A.N.Aheart)

If A, B, C are the angles of a triangle, show that

cosA+ cosB + cosC < 2.

Indeed, cosA+cosB+cosC ≤ 32 , with equality for equilateral triangles.

P.D.Thomas quoted Euler’s relation d2 = R(R− 2r) and

cosA+ cosB + cosC = 1 +r

R

from Johnson (p.191).

David Zeitlin applied the Erdos - Mordell inequality to the circumcenter.

Also,

∑cosA =

32− d2

2R2.

E1402.602.S608. (F.Leuenberger)

Let r denote the radius of the inscribed sphere of a tetrahedron T and letri, I = 1, 2, 3, 4, denote the radii of the exspheres of T which touch one faceof T and the other three faces of T produced. Show that

∑4i=1 ri ≥ 8r, with

equality if and only if T is isosceles.


E1406.603.S609. (M.Goldberg)

Cut an obtuse triangle into the least number of acute triangles.

E1417.605. (R.Hartop)

Given a unit circle with point P on the circumference and a distance d1,0 < d1 < 2. With center at P , construct the circle of radius d1 to cutthe given circle in two points. If d2 is the distance between these points,construct a new circle of radius d2 with center at P and so obtain d3, d4,. . . . Find limn→∞ dn.

E1411.604.S6010. (Shee)

Let ABCDE be any pentagon inscribed in a circle and let P , Q, R, S, T beintersections of the diagonals such that P and Q lie on AC, Q, R on BD,R, S on CE, and S, T on DA. Prove that

AB · BC · CD ·DE ·EAAC · BD · CE ·DA · EB =

AP ·BQ · CR ·DS · ETCP ·DQ ·ER · AS · BT .


Let A′, B′, C ′ (A′′, B′′, C ′′) be the centers of squares described exteriorly(interiorly) on the sides BC, CA, AB of a triangle ABC. Show that theradical center of the circles A(A′), B(B′), C(C ′) (A(A′′), B(B′′), C(C ′′))coincides with the nine-point center of triangle ABC.

E1425.605. (D.J.Newman)

If a square lies within a triangle, prove that the area of the square does notexceed half the area of the triangle.

E1427.607. (F.Leuenberger)

In a triangle, √3(a+ b+ c) ≥ 2(ha + hb + hc),

with equality if and only if the triangle is equilateral.


E1433.608. (A.Oppenheim)

Let P be apoint in the interior of a triangle and let the distances from thevertices of the triangle be x, y, z, and from the sides of the triangle be p, q,r. Show that

xyz ≥ (q + r)(r + p)(p+ q).

E1436.609. (M.K.Shen)

Through the vertices of a given triangle ABC draw straight lines , m, nrespectively, such that n and intersect in D, and m in E, m and n in Finside the triangle and

(a) ABE = BCF = CAD = DEF = 14ABC.

(b) DEF is similar to ABC and has an area equal to a given fractionof ABC.



Advanced Problems, 1950 – 1959


If through the vertices A, B, C, D of a tetrahedron parallel planes are drawncutting a given line L in points A2, B2, C2, D2 and if A1, B1, C1, D1 arethe points in which the lines AA2, BB−2, CC2, DD2 cut the planes BCD,CDA, DAB, ABC, then

AA2

AA1+BB2

BB1+CC2

CC1+DD2

DD1= 2.

AMM4465.51?.S532.(E.I.Gale)

AXBZ is a jointed rhombus with sides of length 4a. (See figure. Forconvenience, a is taken considerably greaer than half the unit). AO and BOare bars of equal length. The fixed centers are O and O′ with OO′ half aunit. Let O′D = 1

2 , FE = 4a, FZ = HG = BE = 2a, HZ = FG = a. Dis an adjustable set screw on bar FZD so that the length ZD can be set atpleasure.

Show that as D moves in a circle about O′, X describes the general conicof eccentricity 1

ZD .


In a triangle ABC, three lines a′, b′ c′ drawn through the vertices A, B, Cdetermine by their intersections a triangle A′B′C ′ and their isogonals a′′,b′′, c′′ determine a triangle A′′B′′C ′′.

(1) Show that the orthic triangles of A′B′C ′ and A′′B′′C ′′ have equalperimeters.

(2) If, further, a′, b′, c′ are equally inclined to AB, BC, CA, showthat the circles A′B′C ′ and A′′B′′C ′′ are symmetric with respect to the linejoining the symmedian point to the circumcenter.


AMM4477.52?.S535.(C.E.Springer)

Show that the area of the Morley triangle of a triangle is

(∏

sin2 A3 )(1 − 4

∑cos2 A

3 + 16∏

cos A3 )∏

sinA ·∏ sin π−A3

times that of the given triangle.See also AMM1943 p.552.

AMM4485.52?.S536.(S.T.Kao)

Through the orthocenter H of a triangle ABC draw any pair of perpendic-ular lines l1 and l2 and let A1, a2; B1, B2; C1, C2 be the respective points ofintersection with the three sides BC, CA, AB. Show that the three poiintsP , Q, R which divide the three segments A1A2, B1B2, C1C2 in the sameratio r lie on a line.

AMM4491.52?.S538.(C.S.Venkataraman)

If ω denotes the Brocard angle of a triangle ABC, prove that(1) the sides are equal when cotω =

√3;

(2) the squares of the lengths of the sides are in arithmetic progressionwhen cotω = 3cotB.


Given three points Ai, i = 1, 2, 3, and a line L cut by the lines AjAk makingangle αi in thepositive sense. Show that the lines drawn through Ai makingangles π − αi, in the positive sense, with L are concurrent at a point P onthe circumcircle of A1A2A3. Further, if the altitudes with Ai make angles βi

with L then the lines through Ai making angles π−βi with L are concurrenton theecircumcircle of A1A2A3 at a point Q diametrically opposite to P .

AMM4530.533.(V.Thebault)

In a tetrahedron ABCD, let A′, B′, C ′, D′ be the feet of the altitudesAA′, BB′, CC ′, DD′. The planes drawn through the midpoints of B′C ′,C ′A′, A′B′, D′A′, D′B′, D′C ′ perpendicular to BC, CA, AB, DA, DB,DC respectively, are concurrent at a poit P , which is the radical center ofthe spheres described with the vertices A, B, C, D as centers and with thealtitudes AA′, BB′, CC ′, DD′ as radii.


AMM4540.535.(J.Gallego-Diaz)

Determine the equaltion of the most general caurve such that the locus ofthe centers of equilateral triangles inscribed in it is the same curve. (Theequilateral hyperbola is a particular case).

AMM4549.537.S549. (R.Oblath)

The Gauss-Newton line of the complete quadrilateral formed by the fourFeuerbach tangents of a triangle is the Euler line of the triangle.

AMM4558.539.(V.F.Ivanoff)

If an octagon l1l2 · · · l8 is inscribed in a conic, then the eight points of inter-section of the sides li and lj, j ≡ i+ 3 mod 8, lie on another conic.


In a triangle ABC, let P be a point having normal coordinates (x, y, z) andconsider the points A′, B′, C ′ with normal coordinates (−x

2 , y, z), (x,−y2 , z)

and (x, y,− z2 ).

(1) The points A, B, C, A′, B′, C ′ lie on one conic S, and there is a onicwith respect ot which the triangles ABC and A′B′C ′ are self polar.

(2) If AP , BP , CP cut BC, CA, AB in A1, B1, C1 and if A′P , B′P ,C ′P cut B′C ′, C ′A′, A′B′ in A′

1, B′1, C

′1, the triangles ABC and A′B′C ′ are

circumscribed about a conic Σ, the points of tangency being A1, B1, C1 andA′

1, B′1, C

′1.

(3) The conics S and Σ have double contact along the common polar ofP with respect to these conics.

AMM4600.547.S5510. (Bankoff)

Vertices A−C and B−D of squares ABCD are joined by quadrants of circles(B) and (C). A semi-circle (O1) is described internally on the diameterBC and a circle (O2) is drawn tangent to the three arcs. Another circle(O3) is drawn tangent to circle (O2) and to arcs AC and BC, and a righttriangle is formed joining O3 and O1 and dropping a perpendicular from O3

upon BC. Successively tangent circles are drawn in the same manner (with(On) tangent to (On−1) and to arcs AC and BC, and right triangles areformed (with O1On for hypotenuse). SShow that the infinitude of trianglesso construced are Pythagorean.


AMM4611.549,551(corrected). (Thebault)

Given five spheres, if one of them is orthogonal to the four others, then thecenters of the four are the vertices of an orthocentric tetrahedron whoseorthocenter coincides with the center of the fifth sphere.


Having given two confocal conics C1 and C2, let M1 on C1 and M2 on C2

be so chosen that the tangents at these points are perpendicular. Show thatthe envelope of M1M2 is another conic having the same foci and havingasymptotes passing through the intersections of C1 and C2.

AMM4651.557.S569. (J.Rosenbaum)

On the sides of a parallelogram A1A2A3A4 , equilateral triangles AiAi+1Bi

are constructed exteriorly. Then equilateral triangles BiBi+1Ci are con-structed interiorly to B1B2B3B4. Prove that C1C2C3C4 coincides withA1A2A3A4.

Generalize to the case when the original parallelogram is replaced by apolygon of n sides.

AMM4669.561.S572. (M.Goldberg)

(1) What is the relation connecting the lengths of the sides of rectangularskew (spatial) pentagon?

(2) Among all possible rectangular skew pentagons, what are the lengthsof the sides of the pentagon in which the ratio of the longest to the shortestis a minimum?

AMM4679.563.S573. (H.Demir)

If A1A2A3A4A5 is a cyclic pentagon and if Ωij denotes the orthopole ofthe line AiAj with respect to the triangle formed by the remaining threevertices, then prove that the ten points Ωij all lie on a circle.


Being given a tetrahedron ABCD and the tetrahedron A1B1C1D1 obtainedby passing planes through A, B, C, D parallelt to the opposite faces ofABCD, show that

PA2 + PB2 + PC2 − 2PD2 − PD21


is a constant independent of the position of point P . Extend this propertyto a skew polygon of n vertices.

AMM4700.567. (J.W.Clawson)

It is well known that the midpoints of the diagonals of a 4-lone lie on astraight line which has been called the Newtonian of the 4-line. It is alsowell known that the Newtonians of the five 4-lines obtained by omitting inturn each of the sides of a 5-line concur in a point which we may call theNewtonian point of the 5-line. Prove

1. that the Newtonian points of the six 5-lines obtained by omitting inturn each of the sides of a 6-line lie on a conic which may be calledthe Newtonian conic of the 6-line;

2. that the Newtonian conics of the seven 6-lines obtained by omitting inturn each of the sides of a 7-line concur in three points, two of whichmay be imaginary.

AMM4710.569. (H.Demir)

Prove that if in a complete quadrangle inscribed in a circle (O) one pairof opposite sides are isotomic lines with respect to a triangle inscribed in(O), then the remaining pairs of opposite sides are also isotomic lines withrespect to the same triangle.

AMM4718.571. (V.F.Ivanoff)

The six points of intersection of a conic and a cubic determine a Pascalhaxagon. Show that the residual six points of intersection of the sides of thehexagon with the cubic form two collinear sets, and the lines determined bythese sets meet on the Pacal line.


If the parallels to the asymptotes of a conic (C), drawn through an arbitrarypoint P of its plane, intersect (C) in P1 and P2, if the perpendiculars to PP1

and PP2 at P1 and P2 intersect in a point O, and if the polar of P withrespect to (C) intersects the conic in M ′

1 and M ′2, then the perpendicular

bisector of segment M ′1M

′2 passes through O.


AMM4730.573. (E.J.F.Primrose)

If a finite set of points in complex 3-dimensional space has the property thatthe line joining any two points of the set passes through a third point of thset, must all points of the set be coplanar?

AMM4908.605. (J.Rainwater)

Consider a triangle abc divided into four smaller triangles, a central one definscribed in abc and three otherson the three sides of def . Show that defcannot have the smallest area of the four unless all four are equal with d, e,f the midpoints of the sides of abc.



Elementary Problems, 1976 – 1991.

AMM.741,p61.()

A generalization of Morley’s theorem.Coxeter-Greitzer p.49,163: Morley’s theorem as a consequence of the

fact that A1A2, B1B2 and C1C2 are concurrent

E1030.52.S74,1110.7510.()

Consider the polygon formed by the internal trisectors of the angles of agiven n−gon, intersecting in neightboring pairs.

(a) Prove that a necessary and sufficient condition that the trisectorpolygon be regular is that the parent n−gon be regular when n ≥ 4.

(b) Prove that the area ratio between the parent and trisector polygonsis always irrational.

Counterexamples are given.

E1073.53,417.()

A polygonal spiral A1A2A3 · · · of unit segments winds counterclockwise andis construcrted in the following manner: Point A1 is at the origin, point A2

is at (1, 0), An−1AnAn+1 = 2πn for all n ≥ 2. Is there a point lying within

the interior of each An−1AnAn+1? If so, what are its coordinates?

E1822.65?.S777.(N.Ucoluk)

Let A,A1 and B,B1 be any two pairs of points in the plane. Consider thelocus of points N such taht the angles ANA1 and BNB1 (with measureshaving absolute values α and β respectively) satisfy the codition α = kβ,where k is a given positive real number.

(a) Determine the differentiability properties of this locus, and(b) when the tangent line exists give a geometric procedure (finite) for

its construction.


E2319.72?.S761.(C.S.Ogilvy)

Find the side of the largest cube that can be wholly contained with a tetra-hedron of side.

E2358.755.()

Let ABC be a triangle. If X is a point on side BC, let AX meet thecircumcircle of ABC again at X ′. Prove or disprove: if XX ′ has maximumlength, then AX lies between the median and the internal angle bisectorissuing from A.

E2401.732.S763.(V.F.Ivanoff)

The exterior angle bisectors of a convex polygon P0 form a polygon P1,whose exterior angle bisectors form a polygon P2, and so on. Prove that Pn

approaches a regular polygon as n→ ∞.

E2407.73?.S74? (A.W.Walker)

Given the circumcenter O, the orthocenter H, and the incenter I of anunknown triangle (T ), (a) locate by euclidean construction the Gergonnepoint and the Lemoine point of (T ), (b) locate the orthocenters of the pedaltriangles of H and I.

Editorial Note: This problem is interesting because triangle (T ) cannotin general be constructed from the given points, but many points related to(T ), including those mentioned in this problem, can be so constructed. Thetwo solutions received are quite involved, so we do not take the space hereto print either of them.

E2453.741.S752.()

Determine all rational numbers r for which 1, cos 2πr and sin 2πr are linearlydependent over the rationals.

E2462.743.S855.(H.Demir)

Let P be an interior point of ABC. Erdos -Mordell inequality:

R1 +R2 +R3 ≥ 2(r1 + r2 + r3).

Prove that the above inequality holds for every point P in the plane of ABCwhen we make the interpretation Ri ≥ 0 always and ri is positive or negative


depending on whether P and A are one the same side of BC or on oppositesides.

Solution by Dodge appeared in CM10.p274-281.

E2471.74.S755.()

Let ma, wa and ha denote the median, angle bisector and altitude to side aof ABC respectively. Show athat

(b+ c)2

4bc≤ ma

wa,

b2 + c2

2bc≤ ma

ha.

When does equality hold?

E2475.74.S756.()

Under what conditions can the four tritangent circles of a triangle be rear-ranged so as to be mutually tangent?

E2477.74.S756.()

A straight line L meets the sides BC,CA and AB of ABC with orthocen-ter H at X, Y , Z respectively. DE is a diameter of the circle ABC ThroughX, Y , Z lines B′C ′, C ′A′ and A′B′ are drawn parallel to AE, BE CE toform a triangle A′B′C ′ oppositively similar to ABC. If D′, E′ H ′ are theimages of D, E, H for this similarity, prove that in general

(a) The lines AA′, BB′ CC ′, DD′, HH ′ concur, so that ABCD andA′B′C ′D′ (ABCH and A′B′C ′H ′) are oppositely similar perspective cyclic(orthocentric) quadrangles;

(b) The lines DH ′ and HD′ meet at the invariant point of the similarlity,and DHD′H ′ is a cyclic quadrangle;

(c) The axis L is perpendicular to DD′ and bisects EE′.

E2489.74?.S765.()

Given ABC, find the locus of all points P not necessarily in the plane ofABC, with the property that the three triangles PAB,PBC,PAB havethe same area.


E2498.74?.S765.(R.E.Smith)

Given triangle ABC, find the locus of all points R (not necessarily in theplane of ABC) with the property that the three angles RAB, RBC andRCA have the same area.

E2501.74.S759.()

Let ABC be a triangle with C ≥ B ≥ A. Show that BIO is a right triangleif and only if a : b : c = 3 : 4 : 5.

E2503.74?.S761.(R.F.Jackson)

A fixed disk C0 of unit radius is centered at (−1, 1). Beginning with thedisk C1, centered at (1, 1) and tangent to the x−Axis and to C0, an infinitechain of disks Ck is constructed, each tangent to the x−axis, to C0, andto Ck−1. Find the sum of their areas.

E2504.74?.S761.(Garfunkel)

Prove or disprove

ha +mb + tc ≤√

32

(a+ b+ c).

See also MG752.Editor’s remark: Lu Ting and Richard Lo obtain the following generaliza-tion:

12≤ ta +mb + tc

a+ b+ c≤

√3

2;

14≤ ha +mb + tc

a+ b+ c≤

√3

2;

38≤ ta +mb +mc

a+ b+ c≤ 1.


Extend the medians of a triangle to meet the circumcircles again, and letthese chords be Ma, Mb, Mc respectively. SHow that

See also E2959.

Ma +Mb +Mc ≥ 43(ma +mb +mc);


Ma +Mb +Mc ≥ 23

√3(a+ b+ c).

When does equality occur ?

E2512.751.S763.()

Let T1 and T2 be two triangles with circumcircles C1 and C2 respectively.Show that if T1 meets T2 then some vertex of T1 lies in (or on) C2 or viceversa. Generalize.

E2513.751.S762.(N.Felsinger)

Let P be a simple (non-self-intersecting) planar polygon. If A is a point inth plane, and if E is an edge of P , then E is viewable from A if for everypoint x of E, the line segment joining A to x contains no point of P otherthan x.

(a) Let A and P be arbitrary. Must some edge of P be viewable from A? Examine the cases of A exterior to P and interior to P separately.

(b) Find sufficient conditions of A in order that some edge of P is view-able from A.

E2514.751.S763.(G.A.Tsintsifas)

Let P be a convex polygon and let K be the polygon whose vertices are themidpoints of the sides of P . A polygon M is formed by dividing the sidesof P (cyclically directed) in a fixed ratio p : q where p+ q = 1. Show that

|M | = (p− q)2|P | + 4pq|K|,

where |M | denotes the area of M etc.

E2517.752.S763.(A.G.Ferrer)

Let P denote a point interior to the triangle ABC, and let r1, r2, r3 denotethe distance from P to the sides of the triangle. If p denotes the perimeterof the pedal triangle, show that

∑(r1 + r2) cos

C

2≤ p.



E2531.754.S766.(V.F.Ivanoff)

Given points A,B,C,D,E, F in the plane, let ABC denote the directed areaof triangle ABC, prove that

AEF ·DBC +BEF ·DCA+ CEF ·DAB = DEF ·ABC.

E2542.741.S752,767.(J.Anderson)

Starting with an arbitrary convex polygon P1, a sequence of polygons isgenerated by successively “chopping off corners”; thus if Pi is a k−gon, thenPi+1 is a (k + 1)−gon, etc. At the jth step, let dj be the altitude of thecut-off triangle, measured from the cut-off vertex. Prove or disprove: Theseries

∑dj converges.

E2553.758.S771.(V.B.Sarma)

Suppose that A, B, C, D are concyclic and that the Simson line of A withrespect to triangle BCD is perpendicular to the Euler line of triangle BCD.Show that the Simsion line of B will be perpendicualr to the Euler lineof triangle CDA. Is the above result true if we replace ‘perpendicular’ by‘parallel’ ?

E2557.758.()

Find all cyclic quadrilaterals with integral sides, each of which has its perime-ter numerically equal to its area.

The following refereences may be of interest E1168[1955,365;1956,43],E2420[1973,691;1974,662]; M.V.Subbarar, Perfect triangles, AMM78(1971),384-385; Marsden, Triangles with integer- valued sides, AMM81(1974),373-376.

E2566.75.S773.(E.Kramer)

A triple of natural numbers is called an obtuse Pythagorean triple ifthey are the sides of a triangle with an angle 120. Such a triple is primitiveif they have no common factor other than 1.

(i) Show that each positive integer except 1, 2, 4, 8 can appear as thesmallest member of an obtuse Pythagorean triple.

(ii)* What positive integers can appear in primitive obtuse Pythagoreantriples?

Answer. (ii): either an odd number > 3 or a multiple of 8. There is ananalogous notion of acute Pythagorean triples (requiring the triangle to


be nonequilateral). If (a, b, c) is an OPT, then (a, c, a + b) is an APT, andall APT can be obtained in this way. Using this it is easy to show that (i)holds also for APT’s.

E2576.752.S775.(R.L.Helmbold)

What is the area of the orthogonal projection of the ellipsoid (xa )2 + (y

b )2 +(z

c )2 = 1 onto a plane perpendicular to the unit vector n = (n1, n2, n3) ?

E2579.762.S776.(B.Klein and B.White)

Let 0 < θ < π2 and let p, q be arbitrary distinct points in the euclidean

plane E. Define fθ(p, q) to be the unique point r in E such that trianglepqr is in the counterclockwise sense and rpq = rqp = θ radians. Showthat fπ/3(p, q) can be written as an expression involving only fπ/6, p, q, andparentheses.

E2584.762.S776.(Coxeter)

Describe an infinite complex congruent isosceles triangles, extending system-atically throughout three-dimensional euclidean space in such a way thateach side of every triangle belongs to just two other triangles.

E2585.762.S776.(J.Mycielski)

Prove that for every triangulation of a 2-dimensional closed surface, theaverage number of edges meeting at a vertex approaches 6 in the limit asthe number of triangles used approaches infinity.

E2617.769.S781.(E.Ehrhart)

A convex body is cut by three parallel planes. If the three sections thusproduced have the same area, show that the portion of the body lying be-tween the two outside plane is a cylinder. Does the same conclusion followif instead we are given that the three sections have the same perimeter ?

E2625.7610.S782.(H.Demir)

Let Ai, i = 0, 1, 2, 3 (mod 4), be four points on a circle Γ. Let ti be thetangent to Γ at Ai and let pi and qi be the lines parallel to ti pasing throughthe points Ai−1 and Ai+1 respectively. If Bi = ti ∩ ti+1, and Ci = pi ∩ qi+1,show that the four lines BiCi have a common point.


E2630.771.S784.(E.T.Ordman)

Suppose that a polyhedral model (made, say, of cardboard) is slit alongcertain edges and unfolded to lie flat in thep lane. The cuts may not bemade so as to disconnect the figure. Now suppose that the resulting planefigure is again folded up to make a polyhedron (folding is allowed only onthe original lines). The new polyhedron is not necessarily congruent to theoriginal one. Find some interesting examples.

E2632.771.S784.(A.Rosenfeld)

Define the discrepancy d(A,B) between two plane geometric figures to be thearea of their symmetric difference. Let A be a circle of radius r. Determinethe inradius of the regular n−gon B for which d(A,B) is minimal.

E2634.771.S784.(Garfunkel)

Let Ai, i ≡ 0, 1, 2 (mod 3), be the vertices of a triangle, Γ its inscribedicrcle with center I. Let Bi be the intersection of the segment AiI of thesegment with Γ and let Ci be the intersection of the line AiI with the sideAi−1Ai. Prove that ∑

AiCi ≤ 3∑

AiBi.


Let ABC be a triangle with A = 40, B = 60. Let D and E be pointslying on the sides AC and AB respectivley, such that CBD = 40 and BCE = 70. Let F be the point where the lines BD and CE intersect.Show that the line AF is perpendicualr to the line BC.

E2641.773.S786.(P.Straffin)

Given a convex polygon, and a point p inside it, define D(p) to be the sum ofperpendicular distance from p to the sides of the polygon (extended if nec-essary). Characgterize those convex polygons for which D(p) is independentof p.

E2646.772.S786.(W.Wernick)

Let A1, . . . , An be vertices of a regular n−gon inscribed in a circle withcenter O. Let B be a point on arc A1An and θ = AnOB. IF ak is thelength of the chord BAk, express

∑nk=1(−1)kak as a function of θ.


E2649.774.S787.(A,Oppenheim)

Let ABC be a non-obtuse triangle, with angles measured in radians. Showthat

(1) 3(a+ b+ c) ≤ π( aA + b

B + cC );

(2) 3(a2 + b2 + c2) ≥ π( a2

A2 + b2

B2 + c2

C2 .

E2657.775.S788.(G.Tsintsifas)

Let A = A0A1 · · ·An and B = B0B1 · · ·Bn be regular n−simplices in Rn.Assume that the ith vertex of B lies on the ith face of A, 0 ≤ i ≤ n. Whatis the minimal value of their similarity ratio ?

E2660.776.S788.(E.Ehrhart)

A quadrilateral is cyclic if its vertices lie on a circle. Find the number ofcongruence classes of cyclic quadrilaterals having integer sides and givenperimeter n.

See also AMM796.p477.

E2668.777.S7810.(R.Evans and I.M.Issacs)

Find all non-isosceles triangles with two or more rational sides and with allangles rational (measured in degrees). Solution. Such a triangle must haveangles 30, 60 and 90.

E2669.777.S7810.(I.J.Schoenberg)

Let a > b > 0. For a given r, 0 < r < b there is a unique R such that thecircle (x−a+r)2+y2 = r2 lies inside and touches the circle x2+(y−b+R)2 =R2. For which r is R

r minimal ?

E2674.778.(G.Tsintsifas)

Let S = A0A1 · · ·An and S′ = A′0A

′1 · · ·A′

n be regular n−simplices such thatA′

i lies on the opposite face of Ai. Is it true that the centroids of S and S′

coincide ?


E2680.779.S792.(J.W.Grossman)

Let ABCD be a convex quadrilateral in the hyperbolic plane. Assume thatAD = BC and that

A+ B = C + D.

Does AB = CD follow from the above hypotheses ? (It does in the euclideanplane).

E2682.779.S793.(D.Hensley)

Let E be an ellipse in the plane whose interior area A ≥ 1. Prove that thenumber n of integer points of E satisfies n < 6 3

√A.

E2687.7710.S799.(R.Evans)

Does there exist a triangle with rational sides whose base equals its altitude? Answer. No.


Let Π be a prism inscribed in a sphere S of unit radius and center O. Thebase of Π is a regular n−gon of radius r. For each face F of Π drop adirected perpendicular from O and let AF be the point where it intersectsS. Let Π∗ be the polyhedron obtained by adding to Π, for each face F , thepyramid of base F and apex AF .

For which values of r is Π∗ convex ?

E2701.783.S795.(R.Stanley)

Find the volume of the convex polytope determined by xi ≥ 0, 1 ≤ i ≤ nand xi + xi+1 ≤ 1, 1 ≤ i ≤ n− 1.

E2715.785.S798,804.(Garfunkel)

Let G be the centroid of ABC. Prove or disprove

sinGAB + sinGBC + sinGCA ≤ 32.

The inequality is true.



Let P be an interior point of triangle ABC. Let A′, B′, C ′ be the pointswhere the perpendiculars drawn from P meet the sides ofABC. LetA′′, B′′, C ′′

be the points where the lines joining P to A,B,C meet the correspondingsides of ABC. Prove or disprove that

A′B′ +B′C ′ + C ′A′ ≤ A′′B′′ +B′′C ′′ +C ′′A′.

E2727.787.S799.(D.P.Robbins)

Two triangles A1A2A3 and B1B2B3 in R3 are equivalent if there exist threedifferent parallel lines p1, p2, p3 and rigid motions σ, τ such that σ(Ai) andτ(Bi) lie on pi, i = 1, 2, 3.

Find necessary and sufficient conditions for equivalence of two triangles.

E2728.787.S799.(J.G.Mauldon)

Let A, b, c, d be radii of four mutually externally tangent right circular cylin-ders whose axes are parallel to the four principal diagonals of a cube. Char-acterize all quadruples a, b, c, d which arise in this way.

E2736.788.S822.(E.Ehrhart)

Let be a closed triangle and P0, A0, P1, A1, . . . an infinite sequence ofpoints in a plane. Assume that Pi = Pi+1, Ai = Ai+1, each Ai is a vertex of and the midpoint of the segment [Pi, Pi+1], and that [Pi, Pi+1]∩ = Ai.Prove that Pn = P0 for some positive n.

E2740.789.S858.(V.Pambuccian)

Show that if P is a convex polyhedron, one can find a square all of whosevertices are on some three faces of P , as well as a square whose vertices areon four different faces of P .

E2746.7810.S801.(G.F.Shum)

Let A1, A2, . . . , An be distinct non-collinear points in the plane. A circlewith center P and radius r is called minimal if AkP ≤ r for all k andequality holds for at least three values of k.

If A1, . . . , An vary, n being fixed, what is the maximum number of min-imal circles ?


S2.791.S802.(Coxeter)

In the hyperbolic plane, the locus of a point at constant distance δ from afixed line (on the side of the line) is one branch of an ‘equidistant’ curve (orhypercycle). In Poincare’s half-plane model, this curve can be representedby a ray making a certain angle with the bounding line of the half-plane.Show that this angle is equal to Π(δ), the angle of parallelism for the distanceδ.

E2751.791.S814.(P.Monsky)

Let X be a conic section. Through what points in space do there pass threemutually perpendicular lines, all meeting X ?

E2757.792.(H.D.Ruderman)

Let a, b, c be three lines in R3. Find points A,B,C on a, b, c respectivelysuch that AB +BC + CA is a minimum.

S12.795.S807.(Klamkin)

If a, a1; b, b1; c, c1 denote the lengths of three pairs of opposite sides of anarbitrary tetrahedron, prove that a + a1, b + b1, c + c1 satisfy the triangleinequality.

S16.797.(I.J.Schoenberg)

Characterize the closed sets S of the complex plane such that d(z + w) ≤d(z) + d(w) for all complex numbers z and w, where d(z) denotes the eu-clidean distance from z to S.

E2790.797.S809.(M.D.Meyerson)

Suppose we have a collection of squares, one each of area 1n for n = 1, 2, 3, . . .

and any open set G in the plane. Show that we can cover all of G except aset of area 0 by placing some of the squares inside G without overlap. (Theedges of the squares are allowed to touch).

S19.798.S812.(Anon, Erewhon-upon-Spanish River∗)

Let C be a smooth simpel arc inside the unit disk, except for its endpoints,which are on the boundary. How long must C be if it cuts off one-third of


the disk’s area ? Generalize.∗ H.Flanders

E2793.798.S819.(E.D.Camier)

P and Q are two points isogonally conjugate with respect to a triangle ABCof which the circumcenter, orthocenter, and nine-point center are O, H, andN respectively. If OR = OP +OQ, and U is the point symmetric to R withrespect to N , show that the isogonal conjugate of U in the triangle ABCis the intersection V of the lines P1Q and PQ1 where P1 and Q1 are theinverses of P and Q in th circle ABC. (Assume that neither of P , Q is onthe circle ABC).

S23.7910.S817.(Garfunkel)

Prove that the sum of the distances from the incenter of a triangle to thevertices does not exceed half of the sum of its internal angle bisectors, eachextended to its intersection with the circumcircle of the triangle.

E2802.799.S811.(M.Slater)

Given a triangle ABC (in the euclidean plane), construct similar isoscelestriangles ABC ′, ACB′ outwards on the respective bases AB and AC, andBCA′′ inwards on the base BC (or ABC ′′ and ACB′′ inwards and BCA′

outwards). Show that AB′A′′C ′ (respectively AB′′A′C ′′) is a parallelogram.

E2816.802.S819.(R.Bojanic)

Consider a circular segment AOB with AOB < π. Let C be the orthogonalprojection of the point B on the line OA. Suppose that the arc AB andthe segment CA are each divided into n equal parts. If M is the point ofpartition of the arc AB closest to B, and N the point of the partition of thesegment CA closest to C, show that the projection of the midpoint of thearc MB onto the line OA is always contained in the interval (C,N).

S29.804.S827.(C.Kimberling)

Suppose T = ABC is a triangle having sides AB < AC < BC and a pointB′ on segment BC satisfying AB′ = AB. Call T admissible if the shortestside of triangle T ′ = AB′C does not touch the shortest side of T , i.e., theshortest side of T ′ is B′C.


(a) Characterize all T for which the sequence T1 = T , T2 = T ′1, T3 = T ′

2,. . . consists exclusively of admissible triangles.

(b) For such T , let sn be the length of the shortest side of Tn anddetermine limn→∞ sn

sn+1.

(c*) For such T , let P be the limit point of the nested triangles Tn anddetermine the angle APB.

E2831.805.S819.(M.Cavachi)

Prove that a convex hexagon with no side longer than 1 unit must have atleast one main diagonal not longer than 2 units.

E2836.807.S825.(J.E.Valentine)

Show that an absolute geometry (no parallel postulate) is euclidean (orriemannian) if some triangle has the property that a median and the segmentjoining the midpoints of the other two sides bisect each other.

E2837.806.S818(C.W.Scherr)

Let aij be the side of a triangle that connects vertices i and j. Let mi be themedian from vertex i. elementary application of the law of cosines yieldsthe relation

at12a

t23a

t31 = λt(at

1 +mt2 +mt

3),

valid for all triangles when t = 2 or t = 4 and λ = 43 . Find an expression

for λ in the limit as t goes to zero. Find the class of triangles for which therelation is valid for a fixed and arbitrary t.

S34.807.S828.(O.Bottema)

In a plane, non-self-intersecting pentagon A1A2A3A4A5 is given. No three ofthe vertices Ai are collinear and (ijk) denotes the signed area of the orientedtriangel AiAjAk. Furthermore,

(124) = a1, (235) = a2, (341) = a3, (452) = a4, (513) = a5.

Determine the area of the pentagon A1A2A3A4A5.The analogous problem, with (123), (234), (345), (451) and (512) being

given, was solved by Gauss in 1823. See Crux 3 (1977), p.240.


E2842.807.S8110.(J.Dou)

Let T be an isosceles right triangle. Let S be the circle such that thedifference between the areas T ∪ S and T ∩ S is minimal. Show that thecenter of S divides the altitude on the hypotenuse of T in the golden ratio.

E2843.807.(P.Ungar)

A set of nonoverlapping rectangles, each having its longer side equal to 1, isinside a circle of diameter

√2. Show that the sum of their area is ≤ 1.

E2848.808.S825.(J.Fickett)

Prove that the regular tetrahedron has minimum diameter among all tetra-hedra that circumscribe a given sphere. (The diameter is the length of alongest edge).

E2866.811.S826.(J.Dou)

Let AKL,AMN be equilateral triangles. Prove that the equilateral trianglesLMX, NKY are concentric (if Y is on the properly chosen side of NK).

E2874.813.S828.(N.Kimura and T.Sekiguchi)

Let n ≥ 3, 0 < Ai ≤ 90. Assume∑n

i=1 cos2Ai = 1. Prove∑tanAi ≥ (n− 1)

∑cotAi.

E2885.815.S844.(T.Sekiguchi)

Let T be a triangle. Construct the set of interior points of T at which thesum of the distances to the sides of T is equal to the arithmetic mean of thelengthes of the altitudes of T .

E2889.815.(I.J.Good)

Let P be an arbitrary point in the plane of a regular polygon A1A2 · · ·An.Let the foot of the perpendicular from P on line AiAi+1 be Qi (where An+1

means A1). Let xi be ± length AiQi: positive if Qi, Ai+1 are on the sameside of Ai; negative otherwise. Prove that

∑xi is equal to half the perimeter

of the polygon.


E2894.816.(T.Ihringer)

Let n be fixed. In how many ways can a square be dissected int (a) ncongruent rectangles, (b) n rectangles of equal area ?

E2905.818.S882.(R.J.Stroeker)

Inside any triangle ABC, a point P exists such that PAB = PBC = PCA := ω. The point P is called a Brocard point and the angle ω is calledits Brocard angle. Prove the inequalities

1ω<

1A

+1B

+1C<

32ω

;3

4ω2<

1A2

+1B2

+1C2

<1ω2.


Let A′, B′, C ′ be the intersection of AI,BI,CI with the incircle of ABC.Continue the process by defining I ′ as the incenter of A′B′C ′, then A′′B′′C ′′

etc. Prove that the angles of A(n)B(n)C(n) approach π3 .

E2911.819.S851.(J.Dou)

Let 2 semicircles AC,CB, AC = 3CB, be given. (A,C,B are collinear). Leta abd b be tangentws to the given semicircles at A, B. Let γ be the circletangent to a and b ant to the larger of the given semicircles. Prove that γ,b and the given semicircles have a common tangent circle. Solution. Invertwith respect to the circle with center at the midpoint of AB and diameterequal in length to CB.

ER2914.8110.S857.(R.C.Lyness)

A circle B lies wholly in the interior of a circle A. S is the set of all circleseach of which touches B externally and A internally.

(i) Find the locus of the internal cneter of similitude of the pairs of circlesfrom S.

(ii) Prove that every point of the locus, except one, is teh internal centerof similitude of exactly one pair of circles from S.


2917.8110.S846.(F.W.Luttman)

Let P0 be a convex polygon of n sides and let 0 < f < 1. Let P0, P1,P2, . . . be a nested sequence of polygons similar to P0 with the followingproperties:

(1) Pk+1 is a linear contraction of Pk by the factor f .(2) Two adjacent sides of Pk+1 lie on Pk. (Necessarily Pk and Pk+1 share

a single vertex).(3) The vertex which Pk shares with Pk+1 lies next clockwise from the

vertex it shares with Pk−1.There is precisely one point lying inside all PK ’s. COnstruct it. (See

Coxeter, Introduction to Geometry, p.164).

E2918.8110.S858.(J.Dou)

Show that an isosceles triangle can be dissected symmetrically around theprincipal median into seven acute isosceles triangles except when the vertexangle is A = 90, 120 or whene 135 ≤ A ≤ 144.


Triangle A1A2A3 is inscribed in a circle; the medians through A1 (A2) meetthe circle again at M1 (M2). The angle bisectors through A1 (A2) meet thecircle again at T1 (T2). Prove or disprove

|A1M1 −A2M2| ≤ |A1T1 −A2T2|.

E2930.822.S842.(Monthy Problem Editors)

Find the largest square that can be inscribed in some triangle of area 1.See also E3114.

E2950.826.S858.(K.W.Lih)

The inner side of a semicircle (including diameter) is a mirror. A light rayemitting from the zenith makes an angle α with the vertical line, 0 ≤ α ≤ π

2 .Characterize α such that the light ray will hit the zenith after finitely manyreflections.

E2958.827.S854.(Klamkin)

Let x, y, z be positive, and let A,B,C be angles of a triangle. Prove thatx2 + y2 + z2 ≥ 2yz sin(A− 30) + 2zx sin(B − 30) + 2xy sin(C − 30).



Triangle ABC is inscribed in a circle. The medians of the triangle intersectat G and are extended to the circle to points D,E,F . Prove that AG +BG+CG ≤ GD +GE +GF .

This is equivalent to part (a) of E2505.

E2962.828.S854.(Klamkin)

It is known that if the circumradii R of the four faces of a tetrahedron arecongruent, then the four facees of the tetrahedron are mutually congruent(i.e., the tetrahedron is isosceles. (See, for example, Crux Math. 6 (1980)219). It is also known that if the inradii r of the four faces of a tetrahedronare congruent, then the tetrahedron need not be isosceles. (See, for example,Crux Math. 4 (1978) 263). Show that if Rr is the same for each face of atetrahedron, the tetrahedron is isosceles.

E2963.828.(C.P.Popescu)

Let A1A2A3, A′1A

′2A

′3 be two equilateral triangles in the plane. Construct

circles γi, γ′i with radii ri (r′i) and centers A − i (A′i), i = 1, 2, 3. Suppose

further that ri (r′i) are geometric progressions with ratio a positive integer.When can the six circles be concurrent ?

E2966.828.S888.(P.J.Giblin)

A,B,P1, P2, P3 are distinct points in the plane. PiPjA, PiPjB are propertriangles, i.e., no two of P1, P2, P3 are collinear with A or with B. Theanticlockwise angles from AP1 to AP2, AP1 to AP3, BP1 to BP2, BP1 toBP3 are θ1, θ2, φ1, φ2. If ai = APi, bi = BPi, and if the relations

sin θ1sinφ1

a3

b3=

sin θ2sinφ2

a2

b2=

sin(θ2 − θ1)sin(φ2 − φ1)

a1

b1

hold, show that the angle APiB has the same pair of bisectors as one of theangles of the triangle P1P2P3. (Possibly the internal bisector of one angle isthe external bisector of the other).

E2967.828.(J.Dou)

Divide a circle into four equiareal parts with (i) arcs (ii) segments of minimaltotal length.



The points A′1, A

′2, A

′3 lie on the sides A2A3, A3A1, A1A2 of an acute angle

triangle A1A2A3 respectively. Show that

2∑

a′i cosAi ≥∑

ai cosAi

where a1, a2, a3 are the sides of the triangle A1A2A3 and a′1, a′2, a′3 are thesides of the triangle A′

1A′2A

′3.

E2974.8210,(Correction 837).S856.(J.Dou)

Let AMB (oriented clockwise) and CMD (counterclockwise) be similar tri-angles. Prove that triangles ACX (clockwise) and Y DB (counterclockwise),both similar to the first triangles, have the same circumcenter.

E2980.831.S926.(J.Dou)

Given the points A1, A2, A3,M and the line s, construct P Q such that PQis equal and parallel to A1M and P1Q1 = P2Q2 = P3Q3, where Pi, Qi arethe intersections of PAi, QAi with s. Describe the locus of the point M forwhich the problem has a solution when A1, A2, A3 and s are known, (fixed).

E2981.831.S864.(Klamkin)

If the three medians of a spherical triangle are equal, must the triangle beequilateral? Note that the sides of a proper spherical triangle are minor arcof great circles and thus its perimeter is < 2π.

E2983.831.S897.(E,Ehrhart)

Let ABC be an equilateral triangle of perimeter 3a. Calculate the area ofthe convex region consisting of all points P such that PA+PB+PC ≤ 2a.


Let Sn = A1A2 · · ·An+1 be an n−simplex in Rn and M a point insides itscircumsphere S : (0, R). The straight line AiM intersects the sphere (0, R)at the point A′

i. We denote K =∑n+1

i=1AiMMA′

i. Let G be the centroid of Sn.

Prove(a) K > n+ 1 if and only if M lies outside the sphere (s) with diameter

OG.


(b) K = n+ 1 if and only if M lies on the sphere (s).(c) K < n+ 1 if and only if M lies in the interior of the sphere (s).

E2990.833.S862.(H.Eves and C.Kimberling)

Let ABC be a triangle and L a line in the plane of ABC not passing throughA, B, C.

(a) Prove that the isogonal conjugate of L is an ellipse, parabola orhyperbola according as L meets the circumcircle of ABC in zero, one ortwo points.

(b) Prove that the isotomic conjugate of L is an ellipse, parabola orhyperbola according as L meets E in zero, one or two points, where E isthe ellipse through A, B, C having the centroid of triangle ABC as center.

E2992.834.(J.Dou)

Find teh shape of a contour of length L that encloses the largest possiblearea and is constrained to pass through three given points.

E2997.835.S864.(I.Adler)

Let p0 be the perimeter of an inscribed regular n-gon in a unit circle, and letdk be the distance from the center of the circle to the side of the inscribedregular 2kn-gon. Prove that

p0

2

∞∏k=1

1dk

= π.

E3007.837.S867.(G.Odom)

Let A and B be the midpoints of the sides EF and ED of an equilateraltriangle DEF . Extend AB to meet the circumcircle (of DEF at C. Showthat B divides AC according to the golden section.

Solution without words.

E3009.837.S8610.(C.Jantzen)

Points X,Y,Z are chosen on the sides of ABC such that

AX

XB=BY

Y C=CZ

ZA= k


and a triangle PQR is formed using CX,AY,BZ as sides. The operation isrepeated on PQR, that is the points X ′.Y ′, Z ′ are chosen on the sides ofPQR such that

PX ′

X ′Q=QY ′

Y ′R=RZ ′

Z ′P= k

and a triangle LMN is formed using RX ′, PY ′, QZ ′ as sides. Show thatLMN is similar to ABC and find the ratio of similarity.

E3013.838.S869.(S.Rabinowitz)

Let ABC be a fixed triangle in the plane. Let T be the transformation of theplane that maps a point P into its isotomic conjgatte (relative to ABC). LetG be the transformation that maps P into its isogonal conjugate. Prove thatthe mappings TG and GT are affine collineations (linear transformations).

E3020.839.S868.(C.Kimberling)

SupposeABC is a nonisosceles triangle. Find three hyperbolas concurrent ina point P such that triangles APB,APC,BPC all have the same perimeter.How does this common perimeter compare with that of ABC?


Prove the inequalities.(a) sinA+ sinB + sinC ≤ 3

√3

2 .(b) sinA sinB sinC ≤ 3

√3

8 .

E3044.845.S871.(J.Dou)

Construct ABC given r,AI,AH.

E3045.845.S874.(C.P.Poposcu)

Let H be a hexagon inscribed in a circle. Show that H can be circumscribedabout a conic if and only if the product of three alternate sides equals theproduct of the other three.

E3049.847.S882.(J.Dou)

Determine a planar region of area 4 which can be partitioned into foursubregions of unit area in such a way that the total length of all bounding


arcs is a minimum. For example, a square of area 4 can be partitioned intofour unit squares so that the bounding arcs have total length 12, while acircle of area 4 can be partitioned into four sectors of unit area so that thetotal length of the bounding arcs is 4π+8√

π≈ 11.6.

E3050.847.(J.Dou)

A optimal tripartition of a plane region is a partition into three subregionsof equal area such that the total length of the separating arcs is a min-imum. Determine those isosceles triangles which admits (a) two optimaltripartitions, (b) two optimal tripartition among those partitions using onlystraight line segments.

E3051.847.(P.O’Hara and H.Sherwood)

It is observed in plane analytic geometry that any set S that is symmetricwith respect to both the x and y axes is also symmetric with respect to theorigin. Does the statement remain valid if the y−axis is replaced by a linethrough the origin with inclination angle α ?

E3054.848.S8610.(V.D.Mascioni)

Prove the inequalities.(a) abcABC ≥ (2π

3 )2r.(b) abc(π −A)(π −B)(π −C) ≥ (4π

√3

9 )3s.

E3059.849.(M.Guan and W.Li)

Let H be a regular n-gon with side length equal to one, n ≥ 4. Show thatif K is any n-gon inscribed in H with side-length xi, i = 1, 2, . . . , n, then

n(1 − cos θ)/2 ≤n∑

i=1

x2i ≤ n(1 − cos θ),

where θ is the internal angle of H. Discuss the case when equality holds.

E3068.8410.S872.(Tsintsifas))

Let T = ABC be a triangle with inradius r and circumradius R. Weconsider a circular disc C with radius d, r ≤ d ≤ R, in a position such thatArea(T ∩C) is a maximum. Prove that as d varies continuously in the closed


interval [r,R], the center of the disc C (in the maximum position) moves ona conic τ passing through the incenter, circumcenter and the Lemoine pointof ABC. Also, ABC is self-polar with respect to the conic τ .

E3071.851.S882.(K.Satyanarayana)

In ABC,A < B < C. Prove that I lies inside OBH.

E3073.852.S882.(C.P.Popescu)

In the hexagon A1A2A3A4A5A6, the triangles A1A3A5 and A2A4A6 areequilateral. Is it true that AkAk+3 are congruent if their sum equals theperimeter of the hexagon? Also, consider the converse.

E3080.853.(L.C.Larson)

Can the following equations be satisfied with integers?

(x+ 1)2 + a2 = (x+ 2)2 + b2 = (x+ 3)2 + c2 = (x+ 4)2 + d2.

E3084.854.S883.(P.Pamfilos)

Given a family of circles in the plane all of which passes through a commonpoint and no two of which are equal, show that there is another circle en-veloping all the circles of the family if and only if there is a straight linecontaining all the intersection points of the common tangents of any twocircles of the family.

E3091.855.S875.(C.P.Popescu)

Equilateral triangle ABC is inscribed in XY Z, with A between Y and Z,B between Z and X, and C between X and Y . Show that

XA+ Y B + ZC < XY + Y Z + ZX.

Is equality possible?

E3098.856.S876.(R.Cuculiere)

Given two circles with diameters IA = a, IB = b, and a set of smaller circlesbetween them as in the following figures, find the total area enclosed by thesmall shaded circles in each of the folowing cases:


(a) The center of one of the small circles lies on the common diameterof the larger circles.

(b) Two of the small circles are tangent to the diameter of the largecircles.

(c) No restrictions.

E3114.859.S881.(M.J.Pelling)

Find the largest cube that can be inscribed in some tetrahedron of volume1.

See also E2930.

E3134.862.S885.(J.Dou)

Construct ABC given ma, wa, A.

E3135.863.S888.(H.Demir)

For a scalene triangle ABC inscribed in a circle, prove that there is a pointD on the arc of the circle opposite to some vertex whose distance from thisvertex is the sum of its distances from the other two vertices. Show how Dmay be constructed with straightedge and compass.

E3146.864.S888.(J.J.Wahl)

Prove or disprove:

2s(√s− a+

√s− b+

√s− c) ≤ 3(

√bc(s − a) +

√ca(s − b) +

√ab(s − c)).

The statement is correct.

E3150.865.S887,9010.(G.A.Tsintsifas)

For arbitrary positive real numbers p, q, r,p

q + ra2 +

q

r + pb2 +

r

p+ qc2 ≥ 2

√3.


Let A1, B1, C1 be points on the sides a, b, c of ABC respectively, anda1, b1, c1 the sides of A1B1C1. Prove that

a2b1c1 + b2c1a1 + c2a1b1 ≥ 42.


E3155.866.S889.(G.Bennet, J.Glenn and C.Kimberling)

Prove that for any ABC, there exist points A′, B′, C ′ such that(1) A′ lies on BC, B′ on AC and C ′ on AB.(2) A′C + CB′ = B′A+AC ′ = C ′B +BA′.(3) AA′, BB′, CC ′ concur in a point.

E3157.866.S886.(L.I.Nicolaescu)

How many sets of four distinct points forming the vertices of a trapezoid arethere if the points are chosen from the vertices of a regular n−gon, n ≥ ?

E3164.867.S887.(H.Demir)

Let s, t be the lengths of the tangent line segments to an ellipse from anexterior point. Find the extreme values of the ratio s

t .

E3167.868.(E.B.Cossi)

Let Pi be any one the five regular polyhedra inscribed in a unit sphere.For each polyhedron Pi, determine the smallest and the largest number ofvertices of Pi which can be seen from a point on a concentric sphere of radiusR > 1.

E3172.869.S8810.8910.(J.Dou)

Let A′ (respectively B′, C ′ be the foot of the altitude from vertex A (respec-tively B, C), in a triangle ABC. Let H be its orthocenter, adn M be anarbitrary point of the plane. Prove that the conics MABA′B′, MBCB′C ′,MCAC ′A′, MHCA′B′, MHAB′C and MHBC ′A′ have a common pointother thatn M .

E3177.8610.S897.(J.Dou)

Let A,B,C be three points on a circle. Let A1 be the intersection of thetangent at A with the line through BC, similarly for B1, C1. Prove that thecircles ABB1, BCC1, CAA1 and the line A1B1C1 have a common point.

E3180.8610.S888.(Klamkin)

Prove thatcos

A

2+ cos

B

2cos

C

≥1 + sinA

2+ sin

B

2sin

C

2.


E3183.871.S892.(Klamkin)

Let P ′ denote the convex n-gon whose vertices are the midpoints of the sidesof a given convex n-gon P . Determine the extreme values of

(i) Area P ′/Area P ,(ii) Perimeter P ′/ Perimeter P .

E3185.871.(R.E.Spaulding)

Let P be a point in the interior of an equilateral triangle, and let S bethe sum of the perpendicular distances to the three sides of the trianglefrom P . In euclidean geometry, the sum S always equals the altitude of thetriangle. In Lobachevskian geometry, prove that S is less than any altitude.In addition, find the position of P which would give a minimum value for S.

E3193.872.S898.(A.Lenard)

Let θ be an undirected acute angle. Show that if a one-to-one mapping Tof the euclidean plane E onto itself has the property that whenever pointsP and Q subtend the angle θ at the point R then also the points T (P )and T (Q) subtend the angle θ at the point T (R), then T is a similaritytransformation of E.

E3195.873.S896.(L.Kuipers)

Given ABC, consider those inscribed ellipses touching AB in C1, BC inA1 and CA in B1 with AB1.BA1 = B1C.A1C. Describe the locus of thecenters of such ellipses.

E3199.873.S897.(Guelicher)

In the triangle ABC, point Q is on the ray BA, point R is on the ray CB,and BQ = CR = AC. A line parallel to AC through R intersects CQ in apoint T . A line parallel to BC through T intersects AC in a point S. Showthat

(AC)3 = AQ ·BC · CS.

E3208.873.S891.()

Suppose that the euclidean plane, line segments of lengths a, b, c, d emanatefrom a given point P in clockwise order, where a, b, c, d are given positive


numbers witha2 + c2 = b2 + d2.

(i) Show that the four segments can be so placed that the endpointsdetermine a rectangle containing P , and show that this rectangle may haveany specified area between 0 and some maximum value M(a, b, c, d).

(ii) Determine M(a, b, c, d).See also MG1147.

E3209.873.S893

Family of circles.

E3231.879.S895.(H.Guelicher)

In a triangle P1P2P3 let pi be the side opposite vertex Pi and let si be a lineparallel to pi (but different from p − I). Suppose that si divides PiPi+1 inthe signed ration λi so that if si meets pi−1 in Qi, then λi = PiQi

QiPi+1. Prove

that the lines s1, s2, s3 are concurrent if and only if

λ1λ2λ3 − (λ1 + λ2 + λ3) = 2.

E3232.879.S896.(J.Dou)

Given lines li, 1 ≤ i ≤ 5 and points Qi, 1 ≤ i ≤ 5, in the plane such thatQi does not lie on li, prove that there exist points Pi, Ri on line li suchthat the angle PiQiRi is a right angle and such that the ten points Pi, Ri,i = 1, 2, . . . , 5 lie on a conic.

E3236.879.S906.(N.D.Rlkies)

For a plane triangle call two circles within the triangle companion incirclesif they are the incircles of the two triangles formed by dividing the giventriangle by passing a line through one of the vertices and some point on theopposite side.

(a) Show that any chain of circles C1, C2, . . . such that Ci and Ci+1 arecompanion incircles for every i consists of at most six distinct circles.

(b) Give a ruler and compass construction for the unique chain whichhas three distinct circles.

(c) For such a chain of three circles show that the three subdividing linesare concurrent.


E3239.8710.(Klamkin)

Show that if A is any three dimensional vector and B, C are unit vectors,then

[(A+B) × (A+ C)] × (B × C) · (B +C) = 0.

Interpret the result as a property of spherical triangles.


Inside a given ABC, it is possible to construct three circles each touchingthe other externally and one side of the triangle. Let D,E,F be the pointsof contact of these circles, appropriately labelled.

(a) Prove that the lines AD,BE,CF are concurrent.(b) Prove that the lines IaD, IbE, IcF are concurrent.

E3256.883.(J.Isbell)

(a) Let T be the set of triangles in the plane whose vertices have integralcoordinates and whose sides have integral lengths. Certain isosceles trianglesin T can be constructed by fitting together two congruent right triangles inT , e.g. the isosceles triangle with vertices (−12,−9), (12, 9), (−12, 16) arisesin this way. Are there any other isosceles triangles in T ?

(b) Consider the set V of triangles in 3-sapce whose vertices have integralcoordinates. Does V contain any equilateral triangles with integral side-length ?

E3257.883.S901.(I.A.Sakmar)

Let P,Q,R be the new vertices of equilateral triangles constructed outwardlyon the edges of a given triangle ABC.

(a) Show that any triangle PQR which can be obtained in this way arisesfrom a unique triangle ABC, and give a construction for recovering triangleABC from triangle PQR.

(b) Show tht not every triangle PQR can be so obtained.

E3259.884.S902.(J.Dou)

Let R be a semicircular region bounded by a line L and a semicircle S withcenter on L. Suppose P1 and P2 are given points in the interior of R. We


wish to find parallel lines l1, l2 through P1, P2 such thatP1C1

P1D1=P2C2

P2D2,

where C1,D1 are the intersections of l1 with S and L and C2, D2 are theintersection of l2 with S and L. Give a necessary and sufficient condition onP1, P2 for such parallel lines to exist.

E3270.886.S899.(B.Lindstrom)

Determine those positive rational numbers m for which arctan√m is a ra-

tional multiple of π.

E3279.887.()

For some n find a simplex in En with integer edges and volume 1.See also MG871.1261.S882

E3282.888.S9010.(D.M.Milosevic)

Prove the inequality

w2a + w2

b + w2c ≤ s2 − r(

R

2− r),


E3283.888.(O.Frink)

Every simple closed polygon in the plane has three centroids, namely thecentroid of its vertex set, the centroid of its boundary, and the centroid ofits interior. In general, all three are distinct.

(a) In the case of a triangle show that these centroids coincide if andonly if the triangle is equilateral.

(b) Which of the three centroids are affine invariant ?

E3293.889.S911.(J.Keane and G.Patruno)

Suppose that the distinct circles C1, C2 intersect at P and Q. Suppose thatthe tangent to C1 at P intersects C2 again at A, the tangent to C2 at Pintersects C1 again at B, and the line AB separates P and Q. Let C3 bethe circle externally tangent to C1, externally tangent to C2, tangent to lineAB, and lying on the same side of AB as Q. Prove that the circles C1 andC2 intercept equal segments on one of the tangents ot C3 through P .


E3299.8810.(J.Yamout)

Suppose ABCD is a plane quadrilateral with no two sides parallel. Let ABand CD intersects at E and AD,BC intersect at F . If M,N,P are themidpoints of AC,BD,EF respectively, and AE = a · AB,AF = b · AD,where a and b are nonzero real numbers, prove that MP = ab ·MN .

E3305.891.S904.()

Determine the maximum values of(i) Q,(ii) |(b− c)(c− a)| + |(c− a)(a− b)| + |(a− b)(b− c)|,(iii) (a− b)2(b− c)2(c− a)2.

E3307.892.S918.()

The celebrated Morley triangle of a given triangle ABC is the equilateraltriangle whose vertices are the intersections of adjacent pairs of internalangles trisectors of ABC. Determine the maximum values of

(i) sM/s,(ii) RM/R,(iii) rM/r,(iv) M/.

E3314.893.S909.()

Let P be a point inside acute triangle ABC. Put α1 = PAC, β1 = PBA, γ1 = PCB. Prove that

cotα1 + cot β1 + cot γ1 > 232 3

14 (cotB + cotB + cotC)

12 .

E3369.902.()

Suppose we are given a piece of paper in the shape of an equilateral triangle.Suppose P is a point in the intersection of the three open circular discs withdiameters AB,BC,CA. If we fold the three corners of the paper in such away that the vertices coincide with P , we get a hexagon three sides of whichare the creases formed and the other three sides of which are portions ofthe sides of ABC. Prove that the area and perimeter of this hexagon areboth maximized when P is the centroid of ABC.


E3375.903

Express

tan∞∑

n=1

arctan1n2

in closed form.

E3377.903.S918.()

Suppose we consider the polygon with vertices e2πiθj , j = 1, 2, . . . , n in thecomplex plane, with

0 ≤ θ1 < θ2 < · · · < θn < 2π.

Suppose α and β are given positive numbers with α+β = 1. Define θ(0)j = θj

for j = 1, 2, · · · , n and

θ(k)j = αθ

(k−1)j + βθ

(k−1)j+1

for k = 1, 2, ... where subscripts are taken modulo n and the values are takenmodulo 2π. Prove that

limk→∞

(θ(k)j+1 − θ

(k)j ) =

2πn,

for j = 1, 2, · · ·.

E3392.906.S922.(A.Bege)

Given an acute triangle ABC with orthocenter H, let A1, B1, C1 be the feetof the altitudes from A,B,C respectively, and let A2, B2, C2 be the feet ofthe perpendicualars fromH onto B1C1, C1A1, A1B1 respectively. Prove that

ABC ≥ 16A2B2C2,

and determine when equality holds.

E3397.907.S921.(J.Chen and C.H.Lo)

The perimeter of ABC is divided into three equal parts by three pointsP,Q,R. Show that

PQR >29ABC,

and that the constant 29 is best possible.



Suppose ABC is a given triangle. Prove the existence of a triangle that isin perspective with every antipedal triangle of ABC.

E3408.909.S922.(J.V.Savall and J.Ferrer)

For each positive integer k, let f(k) denote the number of triangles with inte-ger sides and area k times the perimeter. It is well known (cf. E2420.73?.S74?)that f(1) = 5. Obtain an upper bound for f(k) in terms of k.

The analogous problem for right triangle appeared in CM1447.15?, CMJ232.825.S843.

E3414.9010.S923.(G.Myerson)

Suppose we construct a sequence of rectangles as follows. We begin witha square of area one. We then alternate adjoining a rectangle of area onealongside or on top of the previous rectangle. Find the limiting ratio oflength to height. Answer. π

2 . Beautiful solution by R.M.Robinson.

E3417.911.S927.(R.S.Luthar)

Suppose ABC is a triangle with AB = AC, and let D,E,F,G be points onthe line through B and C defined as follows: D is the midpoint of BNC,AE is the bisector of BAC, F is the foot of the perpeandicualr from A toBC, and AG is perpendicular to AE (i.e. AG bisects one of the exteriorangles at A). Prove that AB · AC = DF ·EG. Solution. Assume b > c.

EG = EB +BG

=ac

b+ c+

ac

b− c

=2abcb2 − c2

=2abc

CF 2 −BF 2

=2abc

(CF +BF )(CF −BF )

=2abc

2a ·DF=

bc

DF.


E3422.912.S927.(H.Demir and C.Tezer)

Suppose F and F ′ are points situated symmetrically with respect to thecenter of a given circle, and suppose S is a point on th circle not on theline FF ′. Let P and P ′ be the second points of intersection of SF andSF ′ respectively with the circle. If the tangents to the circle at P and P ′

intersect at T , prove that the perpendicualr bisector of FF ′ passes throughthe midpoint of the line segment ST .

E3434.914.(John P. Hoyt)

Prove or disprove that there are infinitely many triples of positive integers(a, b, c) with no common factor such that the triangle ABC with sides a, b, chas the following property: the median from A, the angle bisector fromB and the altitude from C are concurrent. Examples of such triples are(12, 13, 15), (35, 277, 308), (26598, 26447, 3193).

See R.K.Guy, My favorite elliptic curve: A tale of two types of triangles,Amer. Math. Monthly, 102 (1995) 771 – 781. See also E374.393.S403, wherethe printed solution is not correct.

E3438.91?S929.(H.Glicher)

Let P1P2P3 have the longest side P1P2. For each of the six permutationsof 1,2,3, let Pij be the point on the ray PiPj such that PkPiPij = PiPjPk.Let pij be the length of PkPij and let pi be the length of PjPk. Prove that

(i) p21 + p2

2 = p23 if and only if p12

p13+ p21

p23= 1;

(ii) p31 + p3


p13+ p32

p23= 1.

E3443.915.S928.(C.P.Popescu)

Let Ai, i = 0, 1, . . . , 5 denote the vertices of a hexagon inscribed in a circleand let Bi denote the intersection of the straight lines AiAi+2 and Ai+1Ai+3

for i = 0, 1, . . . , 5, the indices being computed modulo 6. Prove that if thetriangles A0A2A4 and A1A3A5 have the same orthocenter, then the straightlines BiBi+3, i = 0, 1, 2 are concurrent.

E3450.916.S933.(D.Bowman)

Let T (n) be the number of triangles lying in the subset [0, n] × [0, n] of theplane whose sides lie on lines of slope 0,∞, 1,−1 passing through pointswith integer coordinates. Derive a closed formula for T (n).


E3460.918.S931.(E.Ehrhart)

(a) Suppose we have n mutually perpendicualr chords through a point inte-rior to a sphere S in n−dimensional euclidean space. Prove that the sum ofthe squares of the lengths of these chords depends only on the radius r ofthe sphere and the distance d from P to the center of the sphere.

(b) More generally, suppose 1 ≤ k ≤ n. Each set of k of the n mutuallyperpendicular chords through P given in (a) determines a k−dimensionalaffine subspace. Prove that the sum of the 2

k−powers of the k−dimensional

measure of the cross sections of S made by these

(#1#2

)affine subspaces

depends only on r and d.

E3466.919.S934.(W.Fenton)

Suppose ABC is given. IfX is a point not on any of the lines BC,CA,AB,let the lines AX,BX,CX meet these lines respectively in pointsA′, B′, C ′. Itis known (Miquel’s theorem) that the circles AB′C ′, A′BC ′, A′B′C intersectin a point Y . Prove that X = Y if and only if X is the orthocenter ofABC.

E3468.919.()

Suppose m and n are positive integers such that all primes factors of n arelarger than m.

(a) Prove that

n∑k=1

∗ sin2m(kπ

n) =

φ(n) − µ(n)4m

(#1#2

),

where * denotes summation over integers relatively prime to n.(b) Find a similar formula for cosines.

E3469.9110.S939.(H.Demir)

Suppose P is a point in the interior of ABC, and AP,BP,CP meet thelines BC,CA,AB respectively at the points D,E,F . Prove that the cen-troids of the six triangles PBD,PDC,PCE,PEA,PAF,PFB lie on a conicif and only if P lies on at least one of the three medians of the triangle. o



Advanced Problems, 1976 – 1995.

AMM5499.67?.S68(p.1019),801.(D.E.Daykin)

(1) Are there rational numbers a and b such that no rational number c existfor which there is a triangle having sides a, b, c and area k ?

(2) Prove or disprove the conjecture: for every positive rational numberk for which there exist positive rational a, b, c such that a triangle with sidesa, b, c has area k.

AMM5790.71?.S823.(D.E.Daykin)

Find all nontrivial maps f : R2 → R2 such that whenever a, b, c are collinear,then f(a), f(b), f(c) are collinear.


Let E be the real euclidean plane and 0 < α < 1. What can be said aboutmaps f : E → E which send each triangle T into a triangle fT with areafT ≤ α area T ?

AMM5973.74?S762.(G.Tsintsifas)

Let G = A1, A2, . . . , An be a bounded set of points in the plane. If anythree of these points can be covered by a strip of breadth d, show tht G canbe covered by a strip of breadth 2d.

Find also the minimum real number k, so that any point set G with thegiven property can be covered by a strip of breadth kd.

AMM6062.75?.S777,812,827.(B.H.Voorhees)

Consider an infinite sequence of regular n−gons such that each(n+1)− gonis contained within the preceding n−gon and is of maximal area consistentwith this constraint. Take the first element of this sequence as an equilateraltriangle having unit area. Is the limit of this sequence a point or a circle ?If it is a circle, determine its area.


AMM6089.764.(E.Ehrhart)

Let K be a convex body in Rn of Jordan content V (K) > (n+1)n

n! , and withcentroid at the origin. Does K ∪ (−K) contain a convex body C, symmetricin the origin, for which V (C) > 2n ?

AMM6178.779.S795.(R.Kowalski)

Define the shape of a rectangle to be the ratio of the longer side to theshorter side. Suppose one has an unlimited number of congruent squares atone’s disposal. Given shape α and an error ε, what is the least number ofsquares one needs to construct a rectangle whose shape differs from α byless than ε ?

AMM6179.779.S8010.(E.Ehrhart)

Find all cubes in a cubic lattice whose vertices are lattice points.

AMM6223.787.S799.(H.D.Ruderman)

Let C be a convex curve. Let Q be a convex curve such that the two tangentsto C from each point P of Q form an angle θ fixed in size. Assume that allpoints are in the same plane.

(1) If θ = 90 and Q is a circle, must C be a circle or an ellipse ?(2) If C is an ellipse and θ = 90, what is the nature of Q ?

AMM6298.805.S824.(J.L.Brenner)

If an arbitrary set of 19 lattice points (with integer coordinates) is givenin euclidean 3-space, prove that some three have a centroid with integercoordinates. (This assertion is false if 19 is replaced by 18).

AMM6316.809.S829.(D.Winter)

Let S be a set of 3n points in R3, no four of which are coplanar. Supposethat S = R ∪ Y ∪ G, where each of R, Y , G has n points. Is it possibleto partition S into n triples ri, yi, gi, 1 ≤ i ≤ n, where each ri is in R,each yi is in Y , and each gi is in G, in such a way that the n trianglesTi = convri, yi, gi are pairwise disjoint ?


AMM6322.8010.(M.J.Pelling and Erdos)

Find the largest constant k such that there exists a set S of planar measurek, no three points of which form the vertices of a triangle of area 1. Inparticular, is k = 4π3−

32 ?

AMM6364.819.S835.(K.B.Lsisenring)

A circle with center at the vertex and radius equal to the latus rectum meetsa parabola at P,Q. The circle and parabola have common tangents meetingthe parabola at X,Y . Prove that XP,Y Q are tangent to the circle.

AMM6367.819.S838.(A.Ehrenfeucht and J,Mycielski)

Let A be a finite collection of distinct but possibly overlapping regularn−gons of the same size on the plane such that every vertex of every n−gonof A is a vertex of exactly two n−gons of A.

(a) Construct a collection A of 2n n−gons such that, even if the n−gonsare rigid, A is flexible.

(b) For which n is rigid A possible ?

AMM6377.822.(K.R.Kellum)

Suppose G is a subset of th euclidean plane such that G meets each verticalline in exactly two points and G meets each nonvertical line in a dense setof points. Must G have a subset H such that H meets each vertical line inone point and each nonvertical line in a dense set of points ?

AMM6381.823.(W.W.Meyer)

M.D.Fox (AMM 87 (1980) 708 – 715) defines a Steiner chain as a sequence ofcircles each touching its two neighbours and two given boundary circles CO

and CI . Enlarging on this, with the hypothesis that CO suurounds CI , wedefine a linear chain as a polygon circumscribed by Co and circumscribingCI . Linear or circular, a chain is said to have period n

m if it closes on itself,the first link and the (n + 1)th link coinciding, after m cycles arounds C0.Proe that a linear chain of rational period p and a circular chain of rationalperiod q coexists if and only if

p > 2, q > 2, cosπ

2pcos(

π

4− π

2q) ≤ cos

π

4


and then the boundary circles are uniquely determined as to relative sizeand eccentricity.

AMM6385.824,S839.(M.D.Meyershon)

Prove or disprove: every simple closed curve in euclidean space contains thevertices of a rectangle. (It is known to be true in the euclidean plane).

AMM6388.825.S8310.(N.Wheeler and H.Straubing)

A regular tetrahedron R sits on a unit triangle T on a plane tiled withtriangles congruent to T . A move consists in rotating R about an edge incontact with the plane. After several moves, R sits on T again. Have thevertices of R been permuted in space ? What if R is cube and the tiling isby squares.

AMM6418.831.S848.(G.Benke)

Prove that2N−1∑n=1

sin πn2

2N

sin πn2N

= N.

Original published solution is incorrect.

AMM6477.849.S865;8910. (L.Funar)

Let r be the radius of the incircle of an arbitrary triangle lying in the closedunit square. Prove or disprove that r ≤

√5−14 .


AMM6478.849.(L.Funar)

Let r be the radius of the incircle of an arbitrary triangle lying in a closedfigure F of width w, and let R be the radius of the incircle of F . Are thefollowing inequalities valid ?

(a) 14 ≤ sup r

w ≤ 12 ;

(ii) 12 ≤ sup r

R ≤ 1.


AMM6557.879.S906.(C.Kimberling)

Let C be the circumcircle of ABC. Let A′ be the point, other than A,where the A-median of ABC meets C. Let A′′, be the point, other than A,where the A-altitude of ABC meets C. Similarly define B′, C ′ nd B′′, C ′′.Let DEF be the tangential triangle of ABC (D is the point where theline tangent to C meets the line tangent to C at C). Prove that the linesDA′, EB′, FC ′ and the lines DA′′, EB′′, FC ′′ concur in points that lie onthe Euler line of ABC.

Answer: (a2(b4 + c4 − a4) : · · · : · · ·) and (tanA cos 2A : · · · : · · ·).

AMM6560.8710.(A.J.Krishna,M.M.Rao, and G.S.Rao)

If x and y are odd positive integers, evaluate∞∑

n=1

1n2

tannπ

xtan

nπ

y.

AMM6571.88?.S917.()

Let A(n) be the maximum area of a polygon of n sides of lengths 1, 2, . . . , n,where n ≥ 4. It is known that the maximum area occurs for a polygoninscribed in a circle. (cf. Polya, Mathematics and Plausible Reasoning,vol.1, pp.174-177). Let B(n) denote the area of a regular polygon with nsides and perimeter 1 + 2 + · · · + n. Prove that

1 − A(n)B(n)

π2

3n2.


If k is a poisitive integer, Schinzel, Enseignement Math. (2) 4 (1958) 71 –72, proved that the circle

(X − 1

3

)2

+ Y 2 =

(5k

3

)2

passes through exactly 2k+ 1 lattices points; clearly, the two coordinates ofany one of these 2k + 1 lattice points are of like parity. Thus, by makingthe substitution X = x+ y, Y = x− y, we see that the smaller circle

(x− 1

6

)2

+(y − 1

6

)2

=

(5k√

26

)2

(∗)


also passes through exactly 2k + 1 lattice points.(i) Show that when k = 1 no circle smaller than (*) passes through

exactly 3 lattice points.(ii) Show that when k = 2 no circle smaller than (*) passes through

exactly 5 lattice points.(iii) Show that when 2k + 1 is composite, there is a circle smaller than

(*) which passes through exactly 2k + 1 lattice points.

AMM6628.904.S918.()

Call a triangle a Heron triangle if it has integer sides and integer area.Fermat showed that there does not exist a Heron right triangle whose area isa perfect square. However, the triangle with sides 9, 10, 17 has area 36. Provethat there are infinitely many Heron triangles whose sides have no commonfactor and whose area is a perfect square. Solution. (C.R.Maderer) For eachpositive integer k, define

a(k) := 20k4 + 4k2 + 1,b(k) := 8k6 − 4k4 − 2k2 + 1,c(k) := 8k6 + 8k4 + 10k2.

Here, a(k), b(k) < c(k) < a(k) + b(k). The triangle has area

[(2k)(2k2 − 1)(2k2 + 1)]2.

See also N.J.Fine, On rational triangles, AMM 83 (1976) 517 – 521.(J.Buddenhagen) There are infinitely many pairs of Heron triangles which

share the same square area. Let m > 1 be an odd integer such that 12(m2−1)

is a square. The triangles with sides

12(m3 +m2) − 1,

12(m3 −m2) + 1, m2;

m3 − 12(m− 1), m3 − m+ 1

2, m

both have area 12m

2(m2 − 1).A triangle has rational area if and only if the numbers

t1 =

s(s− a), t2 =

s(s− b)

, t3 =

s(s− c),

are rational. Note that t3 = 1−t1t2t1+t2

. The area of the triangle is a rationalsquare if and only if

u2 = t1t2(t1 + t2)(1 − t1t2).


The conclusion follows from the observation that elliptic curve for t2 = 14

has positive rank.(Editor’s comment): N.Elkies, On A4 +B4 +C4 = D4, Math. Comp/ 51

(1988) 825 – 835, has proved that this equation has infinitely many solutionswith gcd(A,B,C,D) = 1. Since fourth powers are congruent to 1 or 0modulo 16, in such a quadruple, exactly one of A, B, C is odd. If we choosea = B4 +C4, b = A4 +C4, and c = A4 +B4, then gcd(a, b, c) = 1, and a, b,c form a triangle with are (ABCD)2.



Elementary Problems, 1976 – 1991.

AMM.741,p61.()

A generalization of Morley’s theorem.Coxeter-Greitzer p.49,163: Morley’s theorem as a consequence of the

fact that A1A2, B1B2 and C1C2 are concurrent

E1822.65?.S777.(N.Ucoluk)

Let A,A1 and B,B1 be any two pairs of points in the plane. Consider thelocus of points N such that the angles ANA1 and BNB1 (with measureshaving absolute values α and β respectively) satisfy the codition α = kβ,where k is a given positive real number.

(a) Determine the differentiability properties of this locus, and(b) when the tangent line exists give a geometric procedure (finite) for

its construction.

E2319.72?.S761.(C.S.Ogilvy)

Find the side of the largest cube that can be wholly contained with a tetra-hedron of side.

E2358.755.()

Let ABC be a triangle. If X is a point on side BC, let AX meet thecircumcircle of ABC again at X ′. Prove or disprove: if XX ′ has maximumlength, then AX lies between the median and the internal angle bisectorissuing from A.

E2401.732.S763.(V.F.Ivanoff)

The exterior angle bisectors of a convex polygon P0 form a polygon P1,whose exterior angle bisectors form a polygon P2, and so on. Prove that Pn

approaches a regular polygon as n→ ∞.


E2453.741.S752.()

Determine all rational numbers r for which 1, cos 2πr and sin 2πr are linearlydependent over the rationals.

E2462.743.S855.(H.Demir)

Let P be an interior point of ABC. Erdos -Mordell inequality:

R1 +R2 +R3 ≥ 2(r1 + r2 + r3).

Prove that the above inequality holds for every point P in the plane of ABCwhen we make the interpretation Ri ≥ 0 always and ri is positive or negativedepending on whether P and A are one the same side of BC or on oppositesides.

Solution by Dodge appeared in CM10.p274-281.

E2471.74.S755.()

Let ma, wa and ha denote the median, angle bisector and altitude to side aof ABC respectively. Show athat

(b+ c)2

4bc≤ ma

wa,

b2 + c2

2bc≤ ma

ha.


E2475.74.S756.()

Under what conditions can the four tritangent circles of a triangle be rear-ranged so as to be mutually tangent?

E2477.74.S756.()

A straight line L meets the sides BC,CA and AB of ABC with orthocen-ter H at X, Y , Z respectively. DE is a diameter of the circle ABC ThroughX, Y , Z lines B′C ′, C ′A′ and A′B′ are drawn parallel to AE, BE CE toform a triangle A′B′C ′ oppositively similar to ABC. If D′, E′ H ′ are theimages of D, E, H.

(a) The lines AA′, BB′ CC ′, DD′, HH ′ concur, so that ABCD andA′B′C ′D′ (ABCH and A′B′C ′H ′) are oppositely similar perspective cyclic(orthocentric) quadrangles;


(b) The lines DH ′ and HD′ meet at the invariant point of the similarlity,and DHD′H ′ is a cyclic quadrangle;

(c) The axis L is perpendicular to DD′ and bisects EE′.

E2498.74?.S765.(R.E.Smith)

Given triangle ABC, find the locus of all points R (not necessarily in theplane of ABC) with the property that the three triangles RAB, RBC andRCA have the same area.

E2501.74.S759.()

Let ABC be a triangle with C ≥ B ≥ A. Show that BIO is a right triangleif and only if a : b : c = 3 : 4 : 5.

E2503.74?.S761.(R.F.Jackson)

A fixed disk C0 of unit radius is centered at (−1, 1). Beginning with thedisk C1, centered at (1, 1) and tangent to the x−Axis and to C0, an infinitechain of disks Ck is constructed, each tangent to the x−axis, to C0, andto Ck−1. Find the sum of their areas.


Prove or disprove

ha +mb + tc ≤√

32

(a+ b+ c).

See also MG752.Editor’s remark: Lu Ting and Richard Lo obtain the following generaliza-tion:

12≤ ta +mb + tc

a+ b+ c≤

√3

2;

14≤ ha +mb + tc

a+ b+ c≤

√3

2;

38≤ ta +mb +mc

a+ b+ c≤ 1.


Extend the medians of a triangle to meet the circumcircles again, and letthese chords be Ma, Mb, Mc respectively. SHow that


See also E2959.

Ma +Mb +Mc ≥ 43(ma +mb +mc);

Ma +Mb +Mc ≥ 23

√3(a+ b+ c).


E2512.751.S762.(E.A.Herman)

Let T1 and T2 be two triangles with circumcircles C1 and C2 respectively.Show that if T1 meets T2 then some vertex of T1 lies in (or on) C2 or viceversa. Generalize.

E2513.751.S762.(N.Felsinger)

Let P be a simple (non-self-intersecting) planar polygon. If A is a point inthe plane, and if E is an edge of P , then E is viewable from A if for everypoint x of E, the line segment joining A to x contains no point of P otherthan x.

(a) Let A and P be arbitrary. Must some edge of P be viewable from A? Examine the cases of A exterior to P and interior to P separately.

(b) Find sufficient conditions of A in order that some edge of P is view-able from A.


Let P be a convex polygon and let K be the polygon whose vertices are themidpoints of the sides of P . A polygon M is formed by dividing the sidesof P (cyclically directed) in a fixed ratio p : q where p+ q = 1. Show that

|M | = (p− q)2|P | + 4pq|K|,where |M | denotes the area of M etc.

E2517.752.S763.(A.G.Ferrer)

Let P denote a point interior to the triangle ABC, and let r1, r2, r3 denotethe distance from P to the sides of the triangle. If p denotes the perimeterof the pedal triangle, show that

∑(r1 + r2) cos

C

2≤ p.



E2531.754.S766.(V.F.Ivanoff)

Given points A,B,C,D,E, F in the plane, let ABC denote the directed areaof triangle ABC, prove that

AEF ·DBC +BEF ·DCA+ CEF ·DAB = DEF ·ABC.

E2542.741.S752,767.(J.Anderson)

Starting with an arbitrary convex polygon P1, a sequence of polygons isgenerated by successively “chopping off corners”; thus if Pi is a k−gon, thenPi+1 is a (k + 1)−gon, etc. At the jth step, let dj be the altitude of thecut-off triangle, measured from the cut-off vertex. Prove or disprove: Theseries

∑dj converges.

E2553.758.S771.(V.B.Sarma)

Suppose that A, B, C, D are concyclic and that the Simson line of A withrespect to triangle BCD is perpendicular to the Euler line of triangle BCD.Show that the Simsion line of B will be perpendicualr to the Euler lineof triangle CDA. Is the above result true if we replace ‘perpendicular’ by‘parallel’ ?

E2557.758.()

Find all cyclic quadrilaterals with integral sides, each of which has its perime-ter numerically equal to its area.

The following refereences may be of interest E1168[1955,365;1956,43],E2420[1973,691;1974,662]; M.V.Subbarar, Perfect triangles, AMM78(1971),384-385; Marsden, Triangles with integer- valued sides, AMM81(1974),373-376.

E2566.75.S773.(E.Kramer)

A triple of natural numbers is called an obtuse Pythagorean triple ifthey are the sides of a triangle with an angle 120. Such a triple is primitiveif they have no common factor other than 1.

(i) Show that each positive integer except 1, 2, 4, 8 can appear as thesmallest member of an obtuse Pythagorean triple.

(ii)* What positive integers can appear in primitive obtuse Pythagoreantriples?


Answer.. (ii): either an odd number > 3 or a multiple of 8. There is ananalogous notion of acute Pythagorean triples (requiring the triangle tobe nonequilateral). If (a, b, c) is an OPT, then (a, c, a + b) is an APT, andall APT can be obtained in this way. Using this it is easy to show that (i)holds also for APT’s.

E2576.752.S775.(R.L.Helmbold)

What is the area of the orthogonal projection of the ellipsoid (xa )2 + (y

b )2 +(z

c )2 = 1 onto a plane perpendicular to the unit vector n = (n1, n2, n3) ?

E2579.762.S776.(B.Klein and B.White)

Let 0 < θ < π2 and let p, q be arbitrary distinct points in the euclidean

plane E. Define fθ(p, q) to be the unique point r in E such that trianglepqr is in the counterclockwise sense and rpq = rqp = θ radians. Showthat fπ/3(p, q) can be written as an expression involving only fπ/6, p, q, andparentheses.

E2584.762.S776.(Coxeter)

Describe an infinite complex congruent isosceles triangles, extending system-atically throughout three-dimensional euclidean space in such a way thateach side of every triangle belongs to just two other triangles.

E2585.762.S776.(J.Mycielski)

Prove that for every triangulation of a 2-dimensional closed surface, theaverage number of edges meeting at a vertex approaches 6 in the limit asthe number of triangles used approaches infinity.

E2617.769.S781.(E.Ehrhart)

A convex body is cut by three parallel planes. If the three sections thusproduced have the same area, show that the portion of the body lying be-tween the two outside plane is a cylinder. Does the same conclusion followif instead we are given that the three sections have the same perimeter ?

E2625.7610.S782.(H.Demir)

Let Ai, i = 0, 1, 2, 3 (mod 4), be four points on a circle Γ. Let ti be thetangent to Γ at Ai and let pi and qi be the lines parallel to ti pasing through


the points Ai−1 and Ai+1 respectively. If Bi = ti ∩ ti+1, and Ci = pi ∩ qi+1,show that the four lines BiCi have a common point.

E2630.771.S784.(E.T.Ordman)

Suppose that a polyhedral model (made, say, of cardboard) is slit alongcertain edges and unfolded to lie flat in thep lane. The cuts may not bemade so as to disconnect the figure. Now suppose that the resulting planefigure is again folded up to make a polyhedron (folding is allowed only onthe original lines). The new polyhedron is not necessarily congruent to theoriginal one. Find some interesting examples.

E2632.771.S784.(A.Rosenfeld)

Define the discrepancy d(A,B) between two plane geometric figures to be thearea of their symmetric difference. Let A be a circle of radius r. Determinethe inradius of the regular n−gon B for which d(A,B) is minimal.


Let Ai, i ≡ 0, 1, 2 (mod 3), be the vertices of a triangle, Γ its inscribedicrcle with center I. Let Bi be the intersection of the segment AiI of thesegment with Γ and let Ci be the intersection of the line AiI with the sideAi−1Ai. Prove that ∑

AiCi ≤ 3∑

AiBi.


Let ABC be a triangle with A = 40, B = 60. Let D and E be pointslying on the sides AC and AB respectivley, such that CBD = 40 and BCE = 70. Let F be the point where the lines BD and CE intersect.Show that the line AF is perpendicualr to the line BC.

E2641.773.S786.(P.Straffin)

Given a convex polygon, and a point p inside it, define D(p) to be the sum ofperpendicular distance from p to the sides of the polygon (extended if nec-essary). Characgterize those convex polygons for which D(p) is independentof p.


E2646.772.S786.(W.Wernick)

Let A1, . . . , An be vertices of a regular n−gon inscribed in a circle withcenter O. Let B be a point on arc A1An and θ = AnOB. IF ak is thelength of the chord BAk, express

∑nk=1(−1)kak as a function of θ.

E2649.774.S787.(A,Oppenheim)

Let ABC be a non-obtuse triangle, with angles measured in radians. Showthat

(1) 3(a+ b+ c) ≤ π( aA + b

B + cC );

(2) 3(a2 + b2 + c2) ≥ π( a2

A2 + b2

B2 + c2

C2 .


Let A = A0A1 · · ·An and B = B0B1 · · ·Bn be regular n−simplices in Rn.Assume that the ith vertex of B lies on the ith face of A, 0 ≤ i ≤ n. Whatis the minimal value of their similarity ratio ?

E2660.776.S788.(E.Ehrhart)

A quadrilateral is cyclic if its vertices lie on a circle. Find the number ofcongruence classes of cyclic quadrilaterals having integer sides and givenperimeter n.

See also AMM796.p477.

E2668.777.S7810.(R.Evans and I.M.Issacs)

Find all non-isosceles triangles with two or more rational sides and with allangles rational (measured in degrees). Solution. Such a triangle must haveangles 30, 60 and 90.


Let a > b > 0. For a given r, 0 < r < b there is a unique R such that thecircle (x−a+r)2+y2 = r2 lies inside and touches the circle x2+(y−b+R)2 =R2. For which r is R

r minimal ?


E2674.778.(G.Tsintsifas)

Let S = A0A1 · · ·An and S′ = A′0A

′1 · · ·A′

n be regular n−simplices such thatA′

i lies on the opposite face of Ai. Is it true that the centroids of S and S′

coincide ?

E2680.779.S792.(J.W.Grossman)

Let ABCD be a convex quadrilateral in the hyperbolic plane. Assume thatAD = BC and that

A+ B = C + D.

Does AB = CD follow from the above hypotheses ? (It does in the euclideanplane).

E2682.779.S793.(D.Hensley)

Let E be an ellipse in the plane whose interior area A ≥ 1. Prove that thenumber n of integer points of E satisfies n < 6 3

√A.

E2687.7710.S799.(R.Evans)

Does there exist a triangle with rational sides whose base equals its altitude? Answer.. No.


Let Π be a prism inscribed in a sphere S of unit radius and center O. Thebase of Π is a regular n−gon of radius r. For each face F of Π drop adirected perpendicular from O and let AF be the point where it intersectsS. Let Π∗ be the polyhedron obtained by adding to Π, for each face F , thepyramid of base F and apex AF .

For which values of r is Π∗ convex ?

E2701.783.S795.(R.Stanley)

Find the volume of the convex polytope determined by xi ≥ 0, 1 ≤ i ≤ nand xi + xi+1 ≤ 1, 1 ≤ i ≤ n− 1.


E2715.785.S798,804.(Garfunkel)

Let G be the centroid of ABC. Prove or disprove

sinGAB + sinGBC + sinGCA ≤ 32.

The inequality is true.


Let P be an interior point of triangle ABC. Let A′, B′, C ′ be the pointswhere the perpendiculars drawn from P meet the sides ofABC. LetA′′, B′′, C ′′

be the points where the lines joining P to A,B,C meet the correspondingsides of ABC. Prove or disprove that

A′B′ +B′C ′ + C ′A′ ≤ A′′B′′ +B′′C ′′ +C ′′A′.

E2727.787.S799.(D.P.Robbins)

Two triangles A1A2A3 and B1B2B3 in R3 are equivalent if there exist threedifferent parallel lines p1, p2, p3 and rigid motions σ, τ such that σ(Ai) andτ(Bi) lie on pi, i = 1, 2, 3.

Find necessary and sufficient conditions for equivalence of two triangles.

E2728.787.S799.(J.G.Mauldon)

Let A, b, c, d be radii of four mutually externally tangent right circular cylin-ders whose axes are parallel to the four principal diagonals of a cube. Char-acterize all quadruples a, b, c, d which arise in this way.

E2736.788.S822.(E.Ehrhart)

Let be a closed triangle and P0, A0, P1, A1, . . . an infinite sequence ofpoints in a plane. Assume that Pi = Pi+1, Ai = Ai+1, each Ai is a vertex of and the midpoint of the segment [Pi, Pi+1], and that [Pi, Pi+1]∩ = Ai.Prove that Pn = P0 for some positive n.

E2740.789.S858.(V.Pambuccian)

Show that if P is a convex polyhedron, one can find a square all of whosevertices are on some three faces of P , as well as a square whose vertices areon four different faces of P .


E2746.7810.S801.(G.F.Shum)

Let A1, A2, . . . , An be distinct non-collinear points in the plane. A circlewith center P and radius r is called minimal if AkP ≤ r for all k andequality holds for at least three values of k.

If A1, . . . , An vary, n being fixed, what is the maximum number of min-imal circles ?

S2.791.S802.(Coxeter)

In the hyperbolic plane, the locus of a point at constant distance δ from afixed line (on the side of the line) is one branch of an ‘equidistant’ curve (orhypercycle). In Poincare’s half-plane model, this curve can be representedby a ray making a certain angle with the bounding line of the half-plane.Show that this angle is equal to Π(δ), the angle of parallelism for the distanceδ.

E2751.791.S814.(P.Monsky)

Let X be a conic section. Through what points in space do there pass threemutually perpendicular lines, all meeting X ?

E2757.792.(H.D.Ruderman)

Let a, b, c be three lines in R3. Find points A,B,C on a, b, c respectivelysuch that AB +BC + CA is a minimum.

S12.795.S807.(Klamkin)

If a, a1; b, b1; c, c1 denote the lengths of three pairs of opposite sides of anarbitrary tetrahedron, prove that a + a1, b + b1, c + c1 satisfy the triangleinequality.

S16.797.(I.J.Schoenberg)

Characterize the closed sets S of the complex plane such that d(z + w) ≤d(z) + d(w) for all complex numbers z and w, where d(z) denotes the eu-clidean distance from z to S.


E2790.797.S809.(M.D.Meyerson)

Suppose we have a collection of squares, one each of area 1n for n = 1, 2, 3, . . .

and any open set G in the plane. Show that we can cover all of G except aset of area 0 by placing some of the squares inside G without overlap. (Theedges of the squares are allowed to touch).

S19.798.S812.(Anon, Erewhon-upon-Spanish River∗)

Let C be a smooth simpel arc inside the unit disk, except for its endpoints,which are on the boundary. How long must C be if it cuts off one-third ofthe disk’s area ? Generalize.

∗ H.Flanders

E2793.798.S819.(E.D.Camier)

P and Q are two points isogonally conjugate with respect to a triangle ABCof which the circumcenter, orthocenter, and nine-point center are O, H, andN respectively. If OR = OP +OQ, and U is the point symmetric to R withrespect to N , show that the isogonal conjugate of U in the triangle ABCis the intersection V of the lines P1Q and PQ1 where P1 and Q1 are theinverses of P and Q in the circle ABC. (Assume that neither of P , Q is onthe circle ABC).

S23.7910.S817.(Garfunkel)

Prove that the sum of the distances from the incenter of a triangle to thevertices does not exceed half of the sum of its internal angle bisectors, eachextended to its intersection with the circumcircle of the triangle.

E2802.799.S811.(M.Slater)

Given a triangle ABC (in the euclidean plane), construct similar isoscelestriangles ABC ′, ACB′ outwards on the respective bases AB and AC, andBCA′′ inwards on the base BC (or ABC ′′ and ACB′′ inwards and BCA′

outwards). Show that AB′A′′C ′ (respectively AB′′A′C ′′) is a parallelogram.

E2816.802.S819.(R.Bojanic)

Consider a circular segment AOB with AOB < π. Let C be the orthogonalprojection of the point B on the line OA. Suppose that the arc AB andthe segment CA are each divided into n equal parts. If M is the point of


partition of the arc AB closest to B, and N the point of the partition of thesegment CA closest to C, show that the projection of the midpoint of thearc MB onto the line OA is always contained in the interval (C,N).

S29.804.S827.(C.Kimberling)

Suppose T = ABC is a triangle having sides AB < AC < BC and a pointB′ on segment BC satisfying AB′ = AB. Call T admissible if the shortestside of triangle T ′ = AB′C does not touch the shortest side of T , i.e., theshortest side of T ′ is B′C.

(a) Characterize all T for which the sequence T1 = T , T2 = T ′1, T3 = T ′

2,. . . consists exclusively of admissible triangles.

(b) For such T , let sn be the length of the shortest side of Tn anddetermine limn→∞ sn

sn+1.

(c*) For such T , let P be the limit point of the nested triangles Tn anddetermine the angle APB.

E2831.805.S819.(M.Cavachi)

Prove that a convex hexagon with no side longer than 1 unit must have atleast one main diagonal not longer than 2 units.

E2836.807.S825.(J.E.Valentine)

Show that an absolute geometry (no parallel postulate) is euclidean (orriemannian) if some triangle has the property that a median and the segmentjoining the midpoints of the other two sides bisect each other.

E2837.806.S818(C.W.Scherr)

Let aij be the side of a triangle that connects vertices i and j. Let mi be themedian from vertex i. elementary application of the law of cosines yieldsthe relation

at12a

t23a

t31 = λt(at

1 +mt2 +mt

3),

valid for all triangles when t = 2 or t = 4 and λ = 43 . Find an expression

for λ in the limit as t goes to zero. Find the class of triangles for which therelation is valid for a fixed and arbitrary t.


S34.807.S828.(O.Bottema)

In a plane, non-self-intersecting pentagon A1A2A3A4A5 is given. No three ofthe vertices Ai are collinear and (ijk) denotes the signed area of the orientedtriangel AiAjAk. Furthermore,

(124) = a1, (235) = a2, (341) = a3, (452) = a4, (513) = a5.

Determine the area of the pentagon A1A2A3A4A5.The analogous problem, with (123), (234), (345), (451) and (512) being

given, was solved by Gauss in 1823. See Crux 3 (1977), p.240.

E2842.807.S8110.(J.Dou)

Let T be an isosceles right triangle. Let S be the circle such that thedifference between the areas T ∪ S and T ∩ S is minimal. Show that thecenter of S divides the altitude on the hypotenuse of T in the golden ratio.

E2843.807.(P.Ungar)

A set of nonoverlapping rectangles, each having its longer side equal to 1, isinside a circle of diameter

√2. Show that the sum of their area is ≤ 1.

E2848.808.S825.(J.Fickett)

Prove that the regular tetrahedron has minimum diameter among all tetra-hedra that circumscribe a given sphere. (The diameter is the length of alongest edge).

E2866.811.S826.(J.Dou)

Let AKL,AMN be equilateral triangles. Prove that the equilateral trianglesLMX, NKY are concentric (if Y is on the properly chosen side of NK).

E2872.813.S82?*****

E2874.813.S828.(N.Kimura and T.Sekiguchi)

Let n ≥ 3, 0 < Ai ≤ 90. Assume∑n

i=1 cos2Ai = 1. Prove∑tanAi ≥ (n− 1)

∑cotAi.



Let T be a triangle. Construct the set of interior points of T at which thesum of the distances to the sides of T is equal to the arithmetic mean of thelengthes of the altitudes of T .

E2889.815.(I.J.Good)

Let P be an arbitrary point in the plane of a regular polygon A1A2 · · ·An.Let the foot of the perpendicular from P on line AiAi+1 be Qi (where An+1

means A1). Let xi be ± length AiQi: positive if Qi, Ai+1 are on the sameside of Ai; negative otherwise. Prove that

∑xi is equal to half the perimeter

of the polygon.

E2894.816.(T.Ihringer)

Let n be fixed. In how many ways can a square be dissected int (a) ncongruent rectangles, (b) n rectangles of equal area ?

E2905.818.S882.(R.J.Stroeker)

Inside any triangle ABC, a point P exists such that PAB = PBC = PCA := ω. The point P is called a Brocard point and the angle ω is calledits Brocard angle. Prove the inequalities

1ω<

1A

+1B

+1C<

32ω

;3

4ω2<

1A2

+1B2

+1C2

<1ω2.


Let A′, B′, C ′ be the intersection of AI,BI,CI with the incircle of ABC.Continue the process by defining I ′ as the incenter of A′B′C ′, then A′′B′′C ′′

etc. Prove that the angles of A(n)B(n)C(n) approach π3 .

E2911.819.S851.(J.Dou)

Let 2 semicircles AC,CB, AC = 3CB, be given. (A,C,B are collinear). Leta abd b be tangentws to the given semicircles at A, B. Let γ be the circletangent to a and b ant to the larger of the given semicircles. Prove that γ,b and the given semicircles have a common tangent circle. Solution. Invert


with respect to the circle with center at the midpoint of AB and diameterequal in length to CB.

ER2914.8110.S857.(R.C.Lyness)

A circle B lies wholly in the interior of a circle A. S is the set of all circleseach of which touches B externally and A internally.

(i) Find the locus of the internal cneter of similitude of the pairs of circlesfrom S.

(ii) Prove that every point of the locus, except one, is teh internal centerof similitude of exactly one pair of circles from S.

2917.8110.S846.(F.W.Luttman)

Let P0 be a convex polygon of n sides and let 0 < f < 1. Let P0, P1,P2, . . . be a nested sequence of polygons similar to P0 with the followingproperties:

(1) Pk+1 is a linear contraction of Pk by the factor f .(2) Two adjacent sides of Pk+1 lie on Pk. (Necessarily Pk and Pk+1 share

a single vertex).(3) The vertex which Pk shares with Pk+1 lies next clockwise from the

vertex it shares with Pk−1.There is precisely one point lying inside all PK ’s. Construct it. (See

Coxeter, Introduction to Geometry, p.164).

E2918.8110.S858.(J.Dou)

Show that an isosceles triangle can be dissected symmetrically around theprincipal median into seven acute isosceles triangles except when the vertexangle is A = 90, 120 or whene 135 ≤ A ≤ 144.


Triangle A1A2A3 is inscribed in a circle; the medians through A1 (A2) meetthe circle again at M1 (M2). The angle bisectors through A1 (A2) meet thecircle again at T1 (T2). Prove or disprove

|A1M1 −A2M2| ≤ |A1T1 −A2T2|.


E2930.822.S842.(Monthy Problem Editors)

Find the largest square that can be inscribed in some triangle of area 1.See also E3114.

E2950.826.S858.(K.W.Lih)

The inner side of a semicircle (including diameter) is a mirror. A light rayemitting from the zenith makes an angle α with the vertical line, 0 ≤ α ≤ π

2 .Characterize α such that the light ray will hit the zenith after finitely manyreflections.

E2958.827.S854.(Klamkin)

Let x, y, z be positive, and let A,B,C be angles of a triangle. Prove thatx2 + y2 + z2 ≥ 2yz sin(A− 30) + 2zx sin(B − 30) + 2xy sin(C − 30).


Triangle ABC is inscribed in a circle. The medians of the triangle intersectat G and are extended to the circle to points D,E,F . Prove that AG +BG+CG ≤ GD +GE +GF .

This is equivalent to part (a) of E2505.

E2962.828.S854.(Klamkin)

It is known that if the circumradii R of the four faces of a tetrahedron arecongruent, then the four facees of the tetrahedron are mutually congruent(i.e., the tetrahedron is isosceles. (See, for example, Crux Math. 6 (1980)219). It is also known that if the inradii r of the four faces of a tetrahedronare congruent, then the tetrahedron need not be isosceles. (See, for example,Crux Math. 4 (1978) 263). Show that if Rr is the same for each face of atetrahedron, the tetrahedron is isosceles.

E2963.828.(C.P.Popescu)

Let A1A2A3, A′1A

′2A

′3 be two equilateral triangles in the plane. Construct

circles γi, γ′i with radii ri (r′i) and centers A − i (A′i), i = 1, 2, 3. Suppose

further that ri (r′i) are geometric progressions with ratio a positive integer.When can the six circles be concurrent ?


E2966.828.S888.(P.J.Giblin)

A,B,P1, P2, P3 are distinct points in the plane. PiPjA, PiPjB are propertriangles, i.e., no two of P1, P2, P3 are collinear with A or with B. Theanticlockwise angles from AP1 to AP2, AP1 to AP3, BP1 to BP2, BP1 toBP3 are θ1, θ2, φ1, φ2. If ai = APi, bi = BPi, and if the relations

sin θ1sinφ1

a3

b3=

sin θ2sinφ2

a2

b2=

sin(θ2 − θ1)sin(φ2 − φ1)

a1

b1

hold, show that the angle APiB has the same pair of bisectors as one of theangles of the triangle P1P2P3. (Possibly the internal bisector of one angle isthe external bisector of the other).

E2967.828.(J.Dou)

Divide a circle into four equiareal parts with (i) arcs (ii) segments of minimaltotal length.


The points A′1, A

′2, A

′3 lie on the sides A2A3, A3A1, A1A2 of an acute angle

triangle A1A2A3 respectively. Show that

2∑

a′i cosAi ≥∑

ai cosAi

where a1, a2, a3 are the sides of the triangle A1A2A3 and a′1, a′2, a′3 are thesides of the triangle A′

1A′2A

′3.

E2974.8210,(Correction 837).S856.(J.Dou)

Let AMB (oriented clockwise) and CMD (counterclockwise) be similar tri-angles. Prove that triangles ACX (clockwise) and Y DB (counterclockwise),both similar to the first triangles, have the same circumcenter.

E2980.831.S926.(J.Dou)

Given the points A1, A2, A3,M and the line s, construct P Q such that PQis equal and parallel to A1M and P1Q1 = P2Q2 = P3Q3, where Pi, Qi arethe intersections of PAi, QAi with s. Describe the locus of the point M forwhich the problem has a solution when A1, A2, A3 and s are known, (fixed).


E2981.831.S864.(Klamkin)

If the three medians of a spherical triangle are equal, must the triangle beequilateral? Note that the sides of a proper spherical triangle are minor arcof great circles and thus its perimeter is < 2π.

E2983.831.S897.(E,Ehrhart)

Let ABC be an equilateral triangle of perimeter 3a. Calculate the area ofthe convex region consisting of all points P such that PA+PB+PC ≤ 2a.


Let Sn = A1A2 · · ·An+1 be an n−simplex in Rn and M a point insides itscircumsphere S : (0, R). The straight line AiM intersects the sphere (0, R)at the point A′

i. We denote K =∑n+1

i=1AiMMA′

i. Let G be the centroid of Sn.

Prove(a) K > n+ 1 if and only if M lies outside the sphere (s) with diameter

OG.(b) K = n+ 1 if and only if M lies on the sphere (s).(c) K < n+ 1 if and only if M lies in the interior of the sphere (s).

E2990.833.S862.(H.Eves and C.Kimberling)

Let ABC be a triangle and L a line in the plane of ABC not passing throughA, B, C.

(a) Prove that the isogonal conjugate of L is an ellipse, parabola orhyperbola according as L meets the circumcircle of ABC in zero, one ortwo points.

(b) Prove that the isotomic conjugate of L is an ellipse, parabola orhyperbola according as L meets E in zero, one or two points, where E isthe ellipse through A, B, C having the centroid of triangle ABC as center.

E2992.834.(J.Dou)

Find the shape of a contour of length L that encloses the largest possiblearea and is constrained to pass through three given points.

E2997.835.S864.(I.Adler)

Let p0 be the perimeter of an inscribed regular n-gon in a unit circle, and letdk be the distance from the center of the circle to the side of the inscribed


regular 2kn-gon. Prove that

p0

2

∞∏k=1

1dk

= π.

E3007.837.S867.(G.Odom)

Let A and B be the midpoints of the sides EF and ED of an equilateraltriangle DEF . Extend AB to meet the circumcircle (of DEF at C. Showthat B divides AC according to the golden section.

Solution without words.

E3009.837.S8610.(C.Jantzen)

Points X,Y,Z are chosen on the sides of ABC such that

AX

XB=BY

Y C=CZ

ZA= k

and a triangle PQR is formed using CX,AY,BZ as sides. The operation isrepeated on PQR, that is the points X ′.Y ′, Z ′ are chosen on the sides ofPQR such that

PX ′

X ′Q=QY ′

Y ′R=RZ ′

Z ′P= k

and a triangle LMN is formed using RX ′, PY ′, QZ ′ as sides. Show thatLMN is similar to ABC and find the ratio of similarity.


Let ABC be a fixed triangle in the plane. Let T be the transformation of theplane that maps a point P into its isotomic conjgatte (relative to ABC). LetG be the transformation that maps P into its isogonal conjugate. Prove thatthe mappings TG and GT are affine collineations (linear transformations).


SupposeABC is a nonisosceles triangle. Find three hyperbolas concurrent ina point P such that triangles APB,APC,BPC all have the same perimeter.How does this common perimeter compare with that of ABC?



Prove the inequalities.(a) sinA+ sinB + sinC ≤ 3

√3

2 .(b) sinA sinB sinC ≤ 3

√3

8 .

E3044.845.S871.(J.Dou)

Construct ABC given r,AI,AH.

E3045.845.S874.(C.P.Poposcu)

Let H be a hexagon inscribed in a circle. Show that H can be circumscribedabout a conic if and only if the product of three alternate sides equals theproduct of the other three.

E3049.847.S882.(J.Dou)

Determine a planar region of area 4 which can be partitioned into foursubregions of unit area in such a way that the total length of all boundingarcs is a minimum. For example, a square of area 4 can be partitioned intofour unit squares so that the bounding arcs have total length 12, while acircle of area 4 can be partitioned into four sectors of unit area so that thetotal length of the bounding arcs is 4π+8√

π≈ 11.6.

E3050.847.(J.Dou)

A optimal tripartition of a plane region is a partition into three subregionsof equal area such that the total length of the separating arcs is a min-imum. Determine those isosceles triangles which admits (a) two optimaltripartitions, (b) two optimal tripartition among those partitions using onlystraight line segments.

E3051.847.(P.O’Hara and H.Sherwood)

It is observed in plane analytic geometry that any set S that is symmetricwith respect to both the x and y axes is also symmetric with respect to theorigin. Does the statement remain valid if the y−axis is replaced by a linethrough the origin with inclination angle α ?


E3054.848.S8610.(V.D.Mascioni)

Prove the inequalities.(a) abcABC ≥ (2π

3 )2r.(b) abc(π −A)(π −B)(π −C) ≥ (4π

√3

9 )3s.

E3059.849.(M.Guan and W.Li)

Let H be a regular n-gon with side length equal to one, n ≥ 4. Show thatif K is any n-gon inscribed in H with side-length xi, i = 1, 2, . . . , n, then

n(1 − cos θ)/2 ≤n∑

i=1

x2i ≤ n(1 − cos θ),

where θ is the internal angle of H. Discuss the case when equality holds.

E3068.8410.S872.(Tsintsifas))

Let T = ABC be a triangle with inradius r and circumradius R. Weconsider a circular disc C with radius d, r ≤ d ≤ R, in a position such thatArea(T ∩C) is a maximum. Prove that as d varies continuously in the closedinterval [r,R], the center of the disc C (in the maximum position) moves ona conic τ passing through the incenter, circumcenter and the Lemoine pointof ABC. Also, ABC is self-polar with respect to the conic τ .

E3071.851.S882.(K.Satyanarayana)

In ABC,A < B < C. Prove that I lies inside OBH.

E3073.852.S882.(C.P.Popescu)

In the hexagon A1A2A3A4A5A6, the triangles A1A3A5 and A2A4A6 areequilateral. Is it true that AkAk+3 are congruent if their sum equals theperimeter of the hexagon? Also, consider the converse.

E3080.853.(L.C.Larson)

Can the following equations be satisfied with integers?

(x+ 1)2 + a2 = (x+ 2)2 + b2 = (x+ 3)2 + c2 = (x+ 4)2 + d2.


E3084.854.S883.(P.Pamfilos)

Given a family of circles in the plane all of which passes through a commonpoint and no two of which are equal, show that there is another circle en-veloping all the circles of the family if and only if there is a straight linecontaining all the intersection points of the common tangents of any twocircles of the family.

E3091.855.S875.(C.P.Popescu)

Equilateral triangle ABC is inscribed in XY Z, with A between Y and Z,B between Z and X, and C between X and Y . Show that

XA+ Y B + ZC < XY + Y Z + ZX.

Is equality possible?

E3098.856.S876.(R.Cuculiere)

Given two circles with diameters IA = a, IB = b, and a set of smaller circlesbetween them as in the following figures, find the total area enclosed by thesmall shaded circles in each of the folowing cases:

(a) The center of one of the small circles lies on the common diameterof the larger circles.

(b) Two of the small circles are tangent to the diameter of the largecircles.

(c) No restrictions.

E3114.859.S881.(M.J.Pelling)

Find the largest cube that can be inscribed in some tetrahedron of volume1.

See also E2930.

E3134.862.S885.(J.Dou)

Construct ABC given ma, wa, A.

E3135.863.S888.(H.Demir)

For a scalene triangle ABC inscribed in a circle, prove that there is a pointD on the arc of the circle opposite to some vertex whose distance from this


vertex is the sum of its distances from the other two vertices. Show how Dmay be constructed with straightedge and compass.

E3146.864.S888.(J.J.Wahl)

Prove or disprove:

2s(√s− a+

√s− b+

√s− c) ≤ 3(

√bc(s − a) +

√ca(s − b) +

√ab(s − c)).

The statement is correct.

E3150.865.S887,9010.(G.A.Tsintsifas)

For arbitrary positive real numbers p, q, r,

p

q + ra2 +

q

r + pb2 +

r

p+ qc2 ≥ 2

√3.


Let A1, B1, C1 be points on the sides a, b, c of ABC respectively, anda1, b1, c1 the sides of A1B1C1. Prove that

a2b1c1 + b2c1a1 + c2a1b1 ≥ 42.

E3155.866.S889.(G.Bennet, J.Glenn and C.Kimberling)

Prove that for any ABC, there exist points A′, B′, C ′ such that(1) A′ lies on BC, B′ on AC and C ′ on AB.(2) A′C + CB′ = B′A+AC ′ = C ′B +BA′.(3) AA′, BB′, CC ′ concur in a point.

E3157.866.S886,8910.(L.I.Nicolaescu)

How many sets of four distinct points forming the vertices of a trapezoid arethere if the points are chosen from the vertices of a regular n−gon, n ≥ ?

E3164.867.S887.(H.Demir)

Let s, t be the lengths of the tangent line segments to an ellipse from anexterior point. Find the extreme values of the ratio s

t .


E3167.868.S9010.(E.B.Cossi)

Let Pi be any one the five regular polyhedra inscribed in a unit sphere.For each polyhedron Pi, determine the smallest and the largest number ofvertices of Pi which can be seen from a point on a concentric sphere of radiusR > 1.

E3172.869.S8810,8910.(J.Dou)

Let A′ (respectively B′, C ′ be the foot of the altitude from vertex A (respec-tively B, C), in a triangle ABC. Let H be its orthocenter, adn M be anarbitrary point of the plane. Prove that the conics MABA′B′, MBCB′C ′,MCAC ′A′, MHCA′B′, MHAB′C and MHBC ′A′ have a common pointother thatn M .

E3177.8610.S897.(J.Dou)

Let A,B,C be three points on a circle. Let A1 be the intersection of thetangent at A with the line through BC, similarly for B1, C1. Prove that thecircles ABB1, BCC1, CAA1 and the line A1B1C1 have a common point.

E3180.8610.S888.(Klamkin)

Prove thatcos

A

2+ cos

B

2cos

C

≥1 + sinA

2+ sin

B

2sin

C

2.

E3183.871.S892.(Klamkin)

Let P ′ denote the convex n-gon whose vertices are the midpoints of the sidesof a given convex n-gon P . Determine the extreme values of

(i) Area P ′/Area P ,(ii) Perimeter P ′/ Perimeter P .

E3185.871.S898.(R.E.Spaulding)

Let P be a point in the interior of an equilateral triangle, and let S bethe sum of the perpendicular distances to the three sides of the trianglefrom P . In euclidean geometry, the sum S always equals the altitude of thetriangle. In Lobachevskian geometry, prove that S is less than any altitude.In addition, find the position of P which would give a minimum value for S.


E3193.872.S898.(A.Lenard)

Let θ be an undirected acute angle. Show that if a one-to-one mapping Tof the euclidean plane E onto itself has the property that whenever pointsP and Q subtend the angle θ at the point R then also the points T (P )and T (Q) subtend the angle θ at the point T (R), then T is a similaritytransformation of E.

E3195.873.S896.(L.Kuipers)

Given ABC, consider those inscribed ellipses touching AB in C1, BC inA1 and CA in B1 with AB1.BA1 = B1C.A1C. Describe the locus of thecenters of such ellipses.

E3199.873.S897.(Guelicher)

In the triangle ABC, point Q is on the ray BA, point R is on the ray CB,and BQ = CR = AC. A line parallel to AC through R intersects CQ in apoint T . A line parallel to BC through T intersects AC in a point S. Showthat

(AC)3 = AQ ·BC · CS.

E3208.873.S891.(I.D.Berg, R.L.Bishop, and H.G.Diamond)

Suppose that the euclidean plane, line segments of lengths a, b, c, d emanatefrom a given point P in clockwise order, where a, b, c, d are given positivenumbers with

a2 + c2 = b2 + d2.

(i) Show that the four segments can be so placed that the endpointsdetermine a rectangle containing P , and show that this rectangle may haveany specified area between 0 and some maximum value M(a, b, c, d).

(ii) Determine M(a, b, c, d).See also MG1147.

E3209.873.S893.(B.Reznick

Family of circles. Let C0, C1, C2, . . . be the sequence of circles in the Carte-sian plane defined as follows:

(i) C0 is the circle x2 + y2 = 1;(ii) for n = 0, 1, 2, . . ., the circle Cn+1 lies in the upper half - plane and

is tangent to Cn as well as to both branches of the hyperbola x2 − y2 = 1.


Let rn be the radius of Cn. Show that rn is an integer and give a formulafor rn.

E3231.879.S895.(H.Guelicher)

In a triangle P1P2P3 let pi be the side opposite vertex Pi and let si be a lineparallel to pi (but different from p − I). Suppose that si divides PiPi+1 inthe signed ration λi so that if si meets pi−1 in Qi, then λi = PiQi

QiPi+1. Prove

that the lines s1, s2, s3 are concurrent if and only if

λ1λ2λ3 − (λ1 + λ2 + λ3) = 2.

E3232.879.S896.(J.Dou)

Given lines li, 1 ≤ i ≤ 5 and points Qi, 1 ≤ i ≤ 5, in the plane such thatQi does not lie on li, prove that there exist points Pi, Ri on line li suchthat the angle PiQiRi is a right angle and such that the ten points Pi, Ri,i = 1, 2, . . . , 5 lie on a conic.

E3236.879.S906.(N.D.Elkies)

For a plane triangle call two circles within the triangle companion incirclesif they are the incircles of the two triangles formed by dividing the giventriangle by passing a line through one of the vertices and some point on theopposite side.

(a) Show that any chain of circles C1, C2, . . . such that Ci and Ci+1 arecompanion incircles for every i consists of at most six distinct circles.

(b) Give a ruler and compass construction for the unique chain whichhas three distinct circles.

(c) For such a chain of three circles show that the three subdividing linesare concurrent.

See AMM 10780.

E3239.8710.S906.(Klamkin)

Show that if A is any three dimensional vector and B, C are unit vectors,then

[(A+B) × (A+ C)] × (B × C) · (B +C) = 0.

Interpret the result as a property of spherical triangles.



Inside a given ABC, it is possible to construct three circles each touchingthe other externally and one side of the triangle. Let D,E,F be the pointsof contact of these circles, appropriately labelled.

(a) Prove that the lines AD,BE,CF are concurrent.(b) Prove that the lines IaD, IbE, IcF are concurrent.

E3256.883.S901.(J.Isbell)

(a) Let T be the set of triangles in the plane whose vertices have integralcoordinates and whose sides have integral lengths. Certain isosceles trianglesin T can be constructed by fitting together two congruent right triangles inT , e.g. the isosceles triangle with vertices (−12,−9), (12, 9), (−12, 16) arisesin this way. Are there any other isosceles triangles in T ?

(b) Consider the set V of triangles in 3-sapce whose vertices have integralcoordinates. Does V contain any equilateral triangles with integral side-length ?

E3257.883.S911.(I.A.Sakmar)

Let P,Q,R be the new vertices of equilateral triangles constructed outwardlyon the edges of a given triangle ABC.

(a) Show that any triangle PQR which can be obtained in this way arisesfrom a unique triangle ABC, and give a construction for recovering triangleABC from triangle PQR.

(b) Show tht not every triangle PQR can be so obtained.

E3259.884.S902.(J.Dou)

Let R be a semicircular region bounded by a line L and a semicircle S withcenter on L. Suppose P1 and P2 are given points in the interior of R. Wewish to find parallel lines l1, l2 through P1, P2 such that

P1C1

P1D1=P2C2

P2D2,

where C1,D1 are the intersections of l1 with S and L and C2, D2 are theintersection of l2 with S and L. Give a necessary and sufficient condition onP1, P2 for such parallel lines to exist.


E3270.886.S899.(B.Lindstrom)

Determine those positive rational numbers m for which arctan√m is a ra-

tional multiple of π.


For some n find a simplex in En with integer edges and volume 1.See also MG871.1261.S882

E3282.888.S9010.(D.M.Milosevic)

Prove the inequality

w2a + w2

b + w2c ≤ s2 − r(

R

2− r),


E3283.888.S909.(O.Frink)

Every simple closed polygon in the plane has three centroids, namely thecentroid of its vertex set, the centroid of its boundary, and the centroid ofits interior. In general, all three are distinct.

(a) In the case of a triangle show that these centroids coincide if andonly if the triangle is equilateral.

(b) Which of the three centroids are affine invariant ?

E3293.889.S911.(J.Keane and G.Patruno)

Suppose that the distinct circles C1, C2 intersect at P and Q. Suppose thatthe tangent to C1 at P intersects C2 again at A, the tangent to C2 at Pintersects C1 again at B, and the line AB separates P and Q. Let C3 bethe circle externally tangent to C1, externally tangent to C2, tangent to lineAB, and lying on the same side of AB as Q. Prove that the circles C1 andC2 intercept equal segments on one of the tangents ot C3 through P .

E3299.8810.S911.(J.Yamout)

Suppose ABCD is a plane quadrilateral with no two sides parallel. Let ABand CD intersects at E and AD,BC intersect at F . If M,N,P are themidpoints of AC,BD,EF respectively, and AE = a · AB,AF = b · AD,where a and b are nonzero real numbers, prove that MP = ab ·MN .


E3305.891.S904.(Klamkin)

Determine the maximum values of(i) Q,(ii) |(b− c)(c− a)| + |(c− a)(a− b)| + |(a− b)(b− c)|,(iii) (a− b)2(b− c)2(c− a)2.

E3307.892.S918.(P.Andrews, Klamkin, and E.T.H.Wang)

The celebrated Morley triangle of a given triangle ABC is the equilateraltriangle whose vertices are the intersections of adjacent pairs of internalangles trisectors of ABC. Determine the maximum values of

(i) sM/s,(ii) RM/R,(iii) rM/r,(iv) M/.

E3314.893.S909.(A.Bege)

Let P be a point inside acute triangle ABC. Put α1 = PAC, β1 = PBA, γ1 = PCB. Prove that

cotα1 + cot β1 + cot γ1 > 232 3

14 (cotB + cotB + cotC)

12 .

E3337.897.S9010.(Klamkin)

Suppose the two longest edges of tetrahedron are a pair of opposite edges.Prove that the three edges incident to some vertex of the tetrahedron arecongruent to the sides of an acute triangle.

E3369.902.S918.(J.Fukuta)

Suppose we are given a piece of paper in the shape of an equilateral triangle.Suppose P is a point in the intersection of the three open circular discs withdiameters AB,BC,CA. If we fold the three corners of the paper in such away that the vertices coincide with P , we get a hexagon three sides of whichare the creases formed and the other three sides of which are portions ofthe sides of ABC. Prove that the area and perimeter of this hexagon areboth maximized when P is the centroid of ABC.


E3375.903.(R.J.Chapman)

Express

tan∞∑

n=1

arctan1n2

in closed form.

E3377.903.S918.()

Suppose we consider the polygon with vertices e2πiθj , j = 1, 2, . . . , n in thecomplex plane, with

0 ≤ θ1 < θ2 < · · · < θn < 2π.

Suppose α and β are given positive numbers with α+β = 1. Define θ(0)j = θj

for j = 1, 2, · · · , n and

θ(k)j = αθ

(k−1)j + βθ

(k−1)j+1

for k = 1, 2, ... where subscripts are taken modulo n and the values are takenmodulo 2π. Prove that

limk→∞

(θ(k)j+1 − θ

(k)j ) =

2πn,

for j = 1, 2, · · ·.

E3392.906.S922.(A.Bege)

Given an acute triangle ABC with orthocenter H, let A1, B1, C1 be the feetof the altitudes from A,B,C respectively, and let A2, B2, C2 be the feet ofthe perpendicualars fromH onto B1C1, C1A1, A1B1 respectively. Prove that

ABC ≥ 16A2B2C2,

and determine when equality holds.

E3397.907.S921.(J.Chen and C.H.Lo)

The perimeter of ABC is divided into three equal parts by three pointsP,Q,R. Show that

PQR >29ABC,

and that the constant 29 is best possible.



Suppose ABC is a given triangle. Prove the existence of a triangle that isin perspective with every antipedal triangle of ABC.

Answer: The circumcevian triangle of O. Trivially, ABC is also ananswer.

E3408.909.S922.(J.V.Savall and J.Ferrer)

For each positive integer k, let f(k) denote the number of triangles with inte-ger sides and area k times the perimeter. It is well known (cf. E2420.73?.S74?)that f(1) = 5. Obtain an upper bound for f(k) in terms of k.

The analogous problem for right triangle appeared in CM1447.15?, CMJ232.825.S843.

E3414.9010.S923.(G.Myerson)

Suppose we construct a sequence of rectangles as follows. We begin witha square of area one. We then alternate adjoining a rectangle of area onealongside or on top of the previous rectangle. Find the limiting ratio oflength to height. Answer.. π

2 . Beautiful solution by R.M.Robinson.

E3417.911.S927.(R.S.Luthar)

Suppose ABC is a triangle with AB = AC, and let D,E,F,G be points onthe line through B and C defined as follows: D is the midpoint of BNC,AE is the bisector of BAC, F is the foot of the perpeandicualr from A toBC, and AG is perpendicular to AE (i.e. AG bisects one of the exteriorangles at A). Prove that AB · AC = DF ·EG. Solution. Assume b > c.

EG = EB +BG =ac

b+ c+

ac

b− c=

2abcb2 − c2

=2abc

CF 2 −BF 2

=2abc

(CF +BF )(CF −BF )=

2abc2a ·DF =

bc

DF.

E3422.912.S927.(H.Demir and C.Tezer)

Suppose F and F ′ are points situated symmetrically with respect to thecenter of a given circle, and suppose S is a point on the circle not on theline FF ′. Let P and P ′ be the second points of intersection of SF andSF ′ respectively with the circle. If the tangents to the circle at P and P ′

intersect at T , prove that the perpendicualr bisector of FF ′ passes throughthe midpoint of the line segment ST .


E3434.914.(J.P.Hoyt)

Prove or disprove that there are infinitely many triples of positive integers(a, b, c) with no common factor such that the triangle ABC with sides a, b, chas the following property: the median from A, the angle bisector fromB and the altitude from C are concurrent. Examples of such triples are(12, 13, 15), (35, 277, 308), (26598, 26447, 3193).

See CMJ455.913.S923.

E3438.914.S929.(H.Gulicher)

Let P1P2P3 have the longest side P1P2. For each of the six permutationsof 1,2,3, let Pij be the point on the ray PiPj such that PkPiPij = PiPjPk.Let pij be the length of PkPij and let pi be the length of PjPk. Prove that

(i) p21 + p2


p13+ p21

p23= 1;

(ii) p31 + p3


p13+ p32

p23= 1.

E3443.915.S928.(C.P.Popescu)

Let Ai, i = 0, 1, . . . , 5 denote the vertices of a hexagon inscribed in a circleand let Bi denote the intersection of the straight lines AiAi+2 and Ai+1Ai+3

for i = 0, 1, . . . , 5, the indices being computed modulo 6. Prove that if thetriangles A0A2A4 and A1A3A5 have the same orthocenter, then the straightlines BiBi+3, i = 0, 1, 2 are concurrent.

E3450.916.S933.(D.Bowman)

Let T (n) be the number of triangles lying in the subset [0, n] × [0, n] of theplane whose sides lie on lines of slope 0,∞, 1,−1 passing through pointswith integer coordinates. Derive a closed formula for T (n).

E3460.918.S931.(E.Ehrhart)

(a) Suppose we have n mutually perpendicualr chords through a point inte-rior to a sphere S in n−dimensional euclidean space. Prove that the sum ofthe squares of the lengths of these chords depends only on the radius r ofthe sphere and the distance d from P to the center of the sphere.

(b) More generally, suppose 1 ≤ k ≤ n. Each set of k of the n mutuallyperpendicular chords through P given in (a) determines a k−dimensionalaffine subspace. Prove that the sum of the 2

k−powers of the k−dimensional


measure of the cross sections of S made by these

(#1#2

)affine subspaces

depends only on r and d.

E3466.919.S934.(W.Fenton)

Suppose ABC is given. IfX is a point not on any of the lines BC,CA,AB,let the lines AX,BX,CX meet these lines respectively in pointsA′, B′, C ′. Itis known (Miquel’s theorem) that the circles AB′C ′, A′BC ′, A′B′C intersectin a point Y . Prove that X = Y if and only if X is the orthocenter ofABC.

E3468.919.(C.Cooper, R.E.Kennedy, and S.Rabinowitz)

Suppose m and n are positive integers such that all primes factors of n arelarger than m.

(a) Prove that

n∑k=1

∗ sin2m(kπ

n) =

φ(n) − µ(n)4m

(#1#2

),

where * denotes summation over integers relatively prime to n.(b) Find a similar formula for cosines.

E3469.9110.S939.(H.Demir)

Suppose P is a point in the interior of ABC, and AP,BP,CP meet thelines BC,CA,AB respectively at the points D,E,F . Prove that the cen-troids of the six triangles PBD,PDC,PCE,PEA,PAF,PFB lie on a conicif and only if P lies on at least one of the three medians of the triangle.



Advanced Problems, 1976 – 1995.

AMM5499.67?.S68(p.1019),801.(D.E.Daykin)

(1) Are there rational numbers a and b such that no rational number c existfor which there is a triangle having sides a, b, c and area k ?

(2) Prove or disprove the conjecture: for every positive rational numberk for which there exist positive rational a, b, c such that a triangle withsides a, b, c has area k. Solution. (by the Proposer) If > 2, then the threerationals

5k2 − 4k + 4k2 − 4

,k(k2 − 4k + 20)

2(k2 − 4),

k + 22

are positive and form the sides of a triangle with area k. The case 0 < k ≤ 2follows by a change of scale.

This solution was derived by P. and S. Chowla from N.J.Fine’s unpub-lished solution. S.L. Segal has pointed out that it is a special case of Brah-magupta’s triangle of the 7th century.


Find all nontrivial maps f : R2 → R2 such that whenever a, b, c are collinear,then f(a), f(b), f(c) are collinear.


Let E be the real euclidean plane and 0 < α < 1. What can be said aboutmaps f : E → E which send each triangle T into a triangle fT with areafT ≤ α area T ?

AMM5973.74?S762.(G.Tsintsifas)

Let G = A1, A2, . . . , An be a bounded set of points in the plane. If anythree of these points can be covered by a strip of breadth d, show tht G canbe covered by a strip of breadth 2d.

Find also the minimum real number k, so that any point set G with thegiven property can be covered by a strip of breadth kd.


AMM6053.75?.S77?. (R.Finkelstein)

AMM6062.75?.S777,812,827.(B.H.Voorhees)

Consider an infinite sequence of regular n−gons such that each(n+1)− gonis contained within the preceding n−gon and is of maximal area consistentwith this constraint. Take the first element of this sequence as an equilateraltriangle having unit area. Is the limit of this sequence a point or a circle ?If it is a circle, determine its area.


Let K be a convex body in Rn of Jordan content V (K) > (n+1)n

n! , and withcentroid at the origin. Does K ∪ (−K) contain a convex body C, symmetricin the origin, for which V (C) > 2n ?

AMM6178.779.S795.(R.Kowalski)

Define the shape of a rectangle to be the ratio of the longer side to theshorter side. Suppose one has an unlimited number of congruent squares atone’s disposal. Given shape α and an error ε, what is the least number ofsquares one needs to construct a rectangle whose shape differs from α byless than ε ?

AMM6179.779.S8010.(E.Ehrhart)

Find all cubes in a cubic lattice whose vertices are lattice points.

AMM6223.787.S799.(H.D.Ruderman)

Let C be a convex curve. Let Q be a convex curve such that the two tangentsto C from each point P of Q form an angle θ fixed in size. Assume that allpoints are in the same plane.

(1) If θ = 90 and Q is a circle, must C be a circle or an ellipse ?(2) If C is an ellipse and θ = 90, what is the nature of Q ?

AMM6298.805.S824.(J.L.Brenner)

If an arbitrary set of 19 lattice points (with integer coordinates) is givenin euclidean 3-space, prove that some three have a centroid with integercoordinates. (This assertion is false if 19 is replaced by 18).


AMM6316.809.S829.(D.Winter)

Let S be a set of 3n points in R3, no four of which are coplanar. Supposethat S = R ∪ Y ∪ G, where each of R, Y , G has n points. Is it possibleto partition S into n triples ri, yi, gi, 1 ≤ i ≤ n, where each ri is in R,each yi is in Y , and each gi is in G, in such a way that the n trianglesTi = convri, yi, gi are pairwise disjoint ?

AMM6322.8010.(M.J.Pelling and Erdos)

Find the largest constant k such that there exists a set S of planar measurek, no three points of which form the vertices of a triangle of area 1. Inparticular, is k = 4π3−

32 ?

AMM6364.819.S835.(K.B.Lsisenring)

A circle with center at the vertex and radius equal to the latus rectum meetsa parabola at P,Q. The circle and parabola have common tangents meetingthe parabola at X,Y . Prove that XP,Y Q are tangent to the circle.

AMM6367.819.S838.(A.Ehrenfeucht and J,Mycielski)

Let A be a finite collection of distinct but possibly overlapping regularn−gons of the same size on the plane such that every vertex of every n−gonof A is a vertex of exactly two n−gons of A.

(a) Construct a collection A of 2n n−gons such that, even if the n−gonsare rigid, A is flexible.

(b) For which n is rigid A possible ?

AMM6377.822.(K.R.Kellum)

Suppose G is a subset of the euclidean plane such that G meets each verticalline in exactly two points and G meets each nonvertical line in a dense setof points. Must G have a subset H such that H meets each vertical line inone point and each nonvertical line in a dense set of points ?

AMM6381.823.(W.W.Meyer)

M.D.Fox (AMM 87 (1980) 708 – 715) defines a Steiner chain as a sequence ofcircles each touching its two neighbours and two given boundary circles CO

and CI . Enlarging on this, with the hypothesis that CO suurounds CI , wedefine a linear chain as a polygon circumscribed by Co and circumscribing


CI . Linear or circular, a chain is said to have period nm if it closes on itself,

the first link and the (n + 1)th link coinciding, after m cycles arounds C0.Proe that a linear chain of rational period p and a circular chain of rationalperiod q coexists if and only if

p > 2, q > 2, cosπ

2pcos(

π

4− π

2q) ≤ cos

π

4

and then the boundary circles are uniquely determined as to relative sizeand eccentricity.

AMM6385.824,S839.(M.D.Meyershon)

Prove or disprove: every simple closed curve in euclidean space contains thevertices of a rectangle. (It is known to be true in the euclidean plane).

AMM6388.825.S8310.(N.Wheeler and H.Straubing)

A regular tetrahedron R sits on a unit triangle T on a plane tiled withtriangles congruent to T . A move consists in rotating R about an edge incontact with the plane. After several moves, R sits on T again. Have thevertices of R been permuted in space ? What if R is cube and the tiling isby squares.

AMM6477.84?.S86?.8910.()


√5−14 .


AMM6418.831.S848.(G.Benke)

Prove that2N−1∑n=1

sin πn2

2N

sin πn2N

= N.

AMM6477.849.S865,8910.(L.Funar)


√5−14 .


AMM6478.849.S909.(L.Funar)

Let r be the radius of the incircle of an arbitrary triangle lying in a closedfigure F of width w, and let R be the radius of the incircle of F . Are thefollowing inequalities valid ?

(a) 14 ≤ sup r

w ≤ 12 ;

(ii) 12 ≤ sup r

R ≤ 1.

AMM6557.879.S906.(C.Kimberling)

Let C be the circumcircle of ABC. Let A′ be the point, other than A,where the A-median of ABC meets C. Let A′′, be the point, other than A,where the A-altitude of ABC meets C. Similarly define B′, C ′ nd B′′, C ′′.Let DEF be the tangential triangle of ABC (D is the point where theline tangent to C meets the line tangent to C at C). Prove that the linesDA′, EB′, FC ′ and the lines DA′′, EB′′, FC ′′ concur in points that lie onthe Euler line of ABC.

AMM6560.8710.S898.(A.J.Krishna,M.M.Rao, and G.S.Rao)

If x and y are odd positive integers, evaluate∞∑

n=1

1n2

tannπ

xtan

nπ

y.

AMM6571.88?.S917.()

Let A(n) be the maximum area of a polygon of n sides of lengths 1, 2, . . . , n,where n ≥ 4. It is known that the maximum area occurs for a polygoninscribed in a circle. (cf. Polya, Mathematics and Plausible Reasoning,vol.1, pp.174-177). Let B(n) denote the area of a regular polygon with nsides and perimeter 1 + 2 + · · · + n. Prove that

1 − A(n)B(n)

π2

3n2.


If k is a poisitive integer, Schinzel, Enseignement Math. (2) 4 (1958) 71 –72, proved that the circle

(X − 1

3

)2

+ Y 2 =

(5k

3

)2


passes through exactly 2k+ 1 lattices points; clearly, the two coordinates ofany one of these 2k + 1 lattice points are of like parity. Thus, by makingthe substitution X = x+ y, Y = x− y, we see that the smaller circle

(x− 1

6

)2

+(y − 1

6

)2

=

(5k√

26

)2

(∗)

also passes through exactly 2k + 1 lattice points.(i) Show that when k = 1 no circle smaller than (*) passes through

exactly 3 lattice points.(ii) Show that when k = 2 no circle smaller than (*) passes through

exactly 5 lattice points.(iii) Show that when 2k + 1 is composite, there is a circle smaller than

(*) which passes through exactly 2k + 1 lattice points.

AMM6628.904.S918.(R.A.Melter)

Call a triangle a Heron triangle if it has integer sides and integer area.Fermat showed that there does not exist a Heron right triangle whose area isa perfect square. However, the triangle with sides 9, 10, 17 has area 36. Provethat there are infinitely many Heron triangles whose sides have no commonfactor and whose area is a perfect square. Solution. (C.R.Maderer) For eachpositive integer k, define

a(k) := 20k4 + 4k2 + 1,b(k) := 8k6 − 4k4 − 2k2 + 1,c(k) := 8k6 + 8k4 + 10k2.

Here, a(k), b(k) < c(k) < a(k) + b(k). The triangle has area

[(2k)(2k2 − 1)(2k2 + 1)]2.

See also N.J.Fine, On rational triangles, AMM 83 (1976) 517 – 521.(J.Buddenhagen) There are infinitely many pairs of Heron triangles which

share the same square area. Let m > 1 be an odd integer such that 12(m2−1)

is a square. The triangles with sides

12(m3 +m2) − 1,

12(m3 −m2) + 1, m2;

m3 − 12(m− 1), m3 − m+ 1

2, m

both have area 12m

2(m2 − 1).


A triangle has rational area if and only if the numbers

t1 =

s(s− a), t2 =

s(s− b)

, t3 =

s(s− c),

are rational. Note that t3 = 1−t1t2t1+t2

. The area of the triangle is a rationalsquare if and only if

u2 = t1t2(t1 + t2)(1 − t1t2).

The conclusion follows from the observation that elliptic curve for t2 = 14

has positive rank.(Editor’s comment): N.Elkies, On A4 +B4 +C4 = D4, Math. Comp/ 51

(1988) 825 – 835, has proved that this equation has infinitely many solutionswith gcd(A,B,C,D) = 1. Since fourth powers are congruent to 1 or 0modulo 16, in such a quadruple, exactly one of A, B, C is odd. If we choosea = B4 +C4, b = A4 +C4, and c = A4 +B4, then gcd(a, b, c) = 1, and a, b,c form a triangle with are (ABCD)2.



1992-1999

AMM 10202.923.S939.(J.B.Romero Marquez)

Let A′, B′, C ′ be the feet of the altitudes of ABC and let X,Y,Z be thecenters of the circumscribing rectangles of ABC with edges BC,CA,ABrespectively. Prove that XY Z is a dilation of A′B′C ′.

AMM 10244.927.S945.(K.Bromberg and Stan Wagon)

A classical construction of Miquel starts with an n−vertex polygon and apoint P in the plane (not a vertex of the n−gon), and forms another n−gonas follows:

(1) draw the perpendiculars from P to the (extended) sides of the poly-gon;

(2) connect the feet to obtain another n−gon.These steps are then repeated n times (provided that none of the poly-

gons has P as a vertex). The resulting polygon, denote M(P ) is similar tothe initial n−gon.

(a) Given a triangle, construct the point P for which M(P ) is largest.(b) Given a quadrilateral, is there a euclidean construction of the point

P for which M(P ) is largest ?

AMM 10249.928.S948.(O.Yumlu)

Suppose that the inradius of an isosceles triangle and the ratio of the dis-tances from its incenter to its vertices are given. Give a euclidean construc-tion of the triangle. Solution. Let ABC be the desired isosceles trianglewith AB = AC, incenter I and inradius r. Construct an isosceles righttriangle IXY with IX = XY = r and a right angle at X. Let X ′, X ′′ bepoints on the line IX such that IX ′ : IX ′′ = k = the given ratio IA : IB.The line through X ′ parallel to X ′′Y meets the IY at a point Y ′ such thatIY ′ =

√2kr. On the line XY , mark a point P with XP = IY ′ =

√2kr.

Let C, C′ be the circles with center I passing through X and P respectively.Note that the tangents from any point on C′ to C has square length 2k2r2.


Mark a point Q on C with XQ = r, and extend XQ to meet C′ at K. Fi-nally, extend XI to a point A such that IA = XK. The triangle boundedby the tangents to C from A and at X is the desired isosceles triangle.

To justisfy the construction, let α and β be the angles at A and Brespectively. Since IA(IA − r) = KX(KX − QX) = the square length ofthe tangents from K to C = 2k2r2, we have

r

sin α2

·(

r

sin α2

− r

)= 2k2r2,

1 − sinα

2= 2k2 sin2 α

2.

Since α2 + β = π

2 ,

sinα

2= cos β = 1 − 2 sin2 β

2.

From (1) and (2) it follows that sin β2 = k · sin α

2 , and IA = k · IB.

AMM 10251.929.S955. (J.G.Mauldon)

Let C denote the unit cube, and let P be the set of all pairs [a, b] with a andb mutually perpendicular line segments contained in C.

(a) Evaluate supmin|a|, |b| : [a, b] ∈ P.(b) Deduce the area of the largest square, and the volume of the largest

regular octahedron, that fit into C.

AMM 10254.929.S954. (E.Erhart)

The curve traced out by a fixed point of a closed convex curve as that curverolls without slipping along a second curve wil be a called a “roulette”. LetS be the area of one arch of a roulette traced out by an ellipse of area srolling on a straight line. Prove or disprove that S ≥ 3s, with equality onlyif the ellipse is a circle.

AMM 10256.929.S945. (Klamkin)

Let Ai, A′i, i = 1, 2, 3, 4, be the vertices of a rectangular parallelepiped P,

with A′i diametrically opposite to Ai.Let P be any interior point of P. Prove

that

S ≤ 2(PA1 · PA′1 + PA2 · PA′

2 + PA3 · PA′3 + PA4 · PA′

4)

where S denotes the surface area of P.


AMM 10269.9210.S951.(D.M.Bloom)

Prove that there is constant K < 1 with the following property. Let G be aregular (2m + 1)−gon inscribed in the unit circle, and let any point P ∈ Gbe given, then there are distinct vertices V0 and V1 of G such that

|d(P, V0) − d(P, V1)| ≤ K

m.

AMM 10275.931.S951.(Klamkin)

Let A be a regular n−gon with edge length 2. Denote the consecutivevertices by A0, A1, . . . , An−1 and introduce An as a synonym for A0. Let Bbe a regular n−gon inscribed in A with vertices B0, B1, . . . , Bn−1 where Bi

lies on AiAi+1 and |AiBi| = λ < 1 for 0 ≤ i < n. Also let Ci be the pointon AiAi+1 with |AiCi| = αi ≤ λ for 0 ≤ i < n and let C denote the n−gon,also inscribed in A, with vertices C0, C1, . . . , Cn−1.

With P (F) denoting the perimeter of the figure F , prove that P (C) ≥P (B).

AMM 10282.932.S955. (Erdos)

Let A,B,C be the vertices of a triangle inscribed in a unit circle, and let Pbe a point in the interior of the triangle ABC. Show that

PA · PB · PC <3227.

AMM 10293.933.S953. (M.Rosenfeld)

Suppose four distinct lines through the origin in R3 have the peroperty thatthe six acaute angles between pairs of these lines are all equal. Prove thatthis configuration of four lines is isometric either to the diagonals of a cubeor to a configuration of four of the the six diagonals of a regular icosahedron.

AMM 10303.934.S9410.(D.E.Gurarie)

Let a1, a2, . . . , an be positive real numbers.(a) Find necessary and sufficient conditions on these numbers for there

to exist a convex n−gon which admits an inscribed circle and whose sides,in cyclic order, are a1, . . . , an.

(b) Find the radius of the inscribed circle.


AMM 10308.935.S965.(R.Connelly, J.H.Hubbard and W.Whitely)

Suppose that p1, p2, p3, q1, q2, q3 are six points in the plane and that thedistance between pi and qj, i, j = 1, 2, 3, is i + j. Show that the six pointsare collinear.

AMM 10317.936.S9610. (J.B.R.Marquez)

Let ABC be inscribed in a circle C and let A′, B′, C ′ be the midpoints ofthe arcs BC,CA,AB respectively.

(a) Prove that the incenter of ABC is the orthocenter of A′B′C ′.(b) Prove that the pedal triangle of A′B′C ′ is homothetic to ABC.

AMM 10322.937.S967. (Jiang Huanxin)

Let ABCD and AEFG be squares with the common vertex A and differentedge lengths. Let θ = EAD, 0 < θ < π

2 . Suppose that EF and CDintersect at the point P . For which value of θ will AP be perpendicular toCF ?

AMM .10344.939.(E.Ehrhart)

Let S be a regular tetrahedron, and let P ∈ S. Define DV (P ) to be the sumof the distances from P to the vertices of S, and DE(P ) to be the sum ofthe distances from P to the edges of S. Find the maximum and minimumvalues of DE(P )

DV (P ) .


Let D,E,F be distinct points on the sides BC,CA,AB respectively ofABC. Let α = BDF , β = FDA, γ = ADE, and δ = EDC. IfAD,BE and CF are concurrent and α

β = δγ = m = 1, prove that α = δ and

β = γ.


In triangle ABC, find all points P such that the triangle DEF (with D =AP ∩BC, E = BP ∩ CA, F = CP ∩AB) is equilateral.


AMM 10368.943.S978. (E.Alkan)

For each point O on diameter AB of a caircle, perform the following con-struction. Let the perpendicular to AB at O meet the circle at point P .Inscribe circles in the figures bounded by the circle and the lines AB, OP .Let (the) R and S be the points at which the two incircles to the curvilineartriangles AOP and BOP are tangent to the diameter AB. Show that RPSis independent of the position of O.

AMM 10371.943.S972. (E.Y.Stoyanov)

Let B′ and C ′ be points on the sides AB and AC respectively of a giventriangle ABC, and let P be a point on the segment B′C ′. Determine themaximum value of

min[BPB′], [CPC ′][ABC]

where [F ] denotes the area of F .

AMM 10374.943.(D.L.Book)

Given an integer N , characterize the smallest square in the plane containingN lattice points.

See editorial notes on narrative on this problem.

AMM 10378.944.S978. (B.Poonen)

Given that point D is in the interior of ABC and that there are realnumbers a, b, c, d such that AB = ab, AD = ad, BC = bc, BD = bd andCD = cd. Prove that | ABD| + | ACD| = π

3 .

AMM 10386.945.S996.(J.Tabov)

Let a tetrahedron with vertices A1, A2, A3, A4 have altitudes that meet in apoint H. For any point P , let P1, P2, P3, P4 be the feet of the perpendicularsfrom P to the opposite faces of A1, A2, A3, A4 respectively. Prove that thereexist constants a1, a2, a3, a4 such that one has

PH = a1PP1 + a2PP2 + a3PP3 + a4PP4

for all points P .


AMM 10405.948.(H.Glicher)

Let A1A2A3A4A5A6 be a hexagon circumscribed about a conic, and formthe intersections Pi of AiAi+2 with Ai+1Ai+3, i = 1, 2, . . . , 6. Show that thePi are the vertices of a hexagon inscribed in a conic.

AMM 10413.949.S988. (M.Mocanu)

Four disjoint (except for boundary points) equilateral triangles of sides a, b, c,and d are enclosed in a regular hexagon of unit side.

(a) Prove that 3a+ b+ c+ d ≤ 4√

3.(b) When is 3a+ b+ c+ d = 4

√3 ?

(c) Prove or disprove that a+ b+ c+ d ≤ 2√

3.

AMM 10415.949.S979. (Kitchen)

Let A be a triangle whose centroid is at the origin. Choose k ∈ R, anddilate one of the Napoleon triangles of A by a factor of −k and the otherby a factor of k

1−k . Prove that A is simultaneously perspective with bothdilated triangles.

AMM 10418.9410.S983. (R.Satnoianu)

Given that acute triangle ABC, let ha, hb and hc denote the altitudes ands the semiperimeter. Show that

√3 max(ha, hb, hc) ≥ s.

AMM 10440.953. (M.Cavachi)

Show that the euclidean plane cannot be covered with circular disks havingmutually disjoint interiors.

AMM 10453.956.S986. (Klamkin)

Prove that the following two properties of the altitudes of an n−simplex areequivalent:

(a) the altitudes are concurrent;(b) the feet of the altitudes are the orthocenters of their respective faces.


AMM 10455.955.S988. (Z.Franco)

It is easily seen that a parabola can intersect a circle in at most 4 points.(a) Show that there is a number R such that a regular polygon (of any

number of sides) can intersect a parabola in at most R points.(b) Find the smallest R with this property.

AMM 10462.956. (I.Rivin)

Let and ′ be nondegenerate simplices in En, with (n− 1)−dimensionalfaces Fi and F ′

i respectively, (i = 0, 1, 2, . . . , n). Let αij be the dihedralangle between Fi and Fj , and let α′

ij be the dihedral angle between F ′i and

F ′j , (i = j). Prove that if αij ≥ α′

ij with 0 ≤ i < j ≤ n, then and ′ aresimilar.

AMM 10469.957.S989*. (J.Anglesio)

Let P be a point in the interior of the trianlge ABC and let the lines AP ,BP , CP meet the sides BC, CA, AB respectively at the points D, E, F .Let the circles on diameters BC and AD intersect at the points a, a′; thecircles on diameters CA and BE intersect at points b, b′; and the circles ondiameters AB and CF intersect at points c, c′. Show that a, a′, b, b′, c, c′

lie on a circle.

AMM 10472.957. (E.Kitchen)

Let P0P1P2P3P4 be a convex pentagon that is affinely equivalent to a regularpentagon. Let Lj be the center of a rotation through +π

5 radians taking Pj+2

to Pj−2 (all subscripts modulo 5). Show that Pj is the center of a rotationthrough −3π

5 radians taking Lj−1 to Lj+1.

AMM 10478.958. (J.P.Hutchison)

Let P be a simple closed n−gon, not necessarily convex (an “art galley”),with some pairs of vertices joined by nonintersecting interior diagonals (“walls”),and suppose that in the interior of each of these diagonals there is an arbi-trarily placed, arbitrarily small opening (a “doorway”). Determine the sizeof the smallest set G of points (“guards”) so that every other point q in Pthere is a line segment in P , disjoint from the punctured diagonals, thatjoins q to a point of G.


Proposal for AMM

Let X,Y,Z be the projections of the incenter of ABC on the sides BC,CAand AB respectively. Let X ′, Y ′, Z ′ be the points on the incircle diametri-cally opposite to X,Y,Z respectively. Show that the lines AX ′, BY ′, CZ ′

are concurrent. Solution. The line AX ′, when produced, intersects the sideBC at the point of contact with the excircle on this side. Similarly, for BY ′

and CZ ′. It follows that these three lines intersect at the Nagel point ofthe triangle.

To justify the first statement, let P be the point onBC so thatBP = s−cand CP = s − b. This is the point of contact of the side BC with theexcircle. Let I be the incenter, Ia the excenter on the side BC, and ra thecorresponding exradius. Produce XI to meet AP at X ′′. It follows that

IX ′′

IaP=

IA

IaA=

r

ra.

Since IaP = ra, we have IX ′′ = r and X ′′ = X ′, the point on the incirclediametrically opposite to X.

With this observation, the concurrency of the lines AX ′, BY ′, CZ ′ fol-lows from Ceva’s theorem.

AMM 10474.958. (H.Tamvikas)

Consider a triangle ABC and a point P in the interior of ABC, and letthe lines AP , BP , CP meet the lines BC, CA, AB at the points D, E, Frespectively. Show that EDF is a right angle if and only if

1|DP | =

1|AD| +

1|BD| +

1|CD| .

Solution. We first establish the equivalence of the following statements.

(i) DE bisects ADC.(ii) DF bisects ADB.(iii) EDF is a right angle.

Suppose DE bisects ADC, so that in triangle ADC, |CE||AE| = |CD|

|AD| .Applying Menelau’s theorem to triangle ADC with transversal BPE,we have |CE|

|AE| ·|AP ||DP | ·

|BD||BC| = 1.


This means |BC||BD| =

|AP ||DP | ·

|CE||AE| =

|AP ||DP | ·

|CD||AD| . (2)

This can be reorganized as

|BC||CD| =

|AP ||DP | ·

|BD||AD| , (3)

from which we infer |BF ||AF | = |BD|

|AD| , and DF bisects ADB. This proves that(i) implies (ii). Similarly, (ii) implies (i). The equivalence of (i) and (ii)clearly shows that each one of them implies (iii). That (iii) implies (i) and(ii) is a consequence of the more general fact: in triangle ADC, CDE isgreater than, equal to, or less than ADE according as |CE|

|AE| is greater than,

equal to, or less than |CD||AD| . Also, in triangle ADB, BDF is greater than,

equal to, or less than ADF according as |BF ||AF | is greater than, equal to, or

less than |BD||AD| .

Now, suppose any of (i), (ii), (iii) holds. Then from (1) and (2), we have

|BC||BD| +

|BC||CD| =

|AP ||DP |

( |CD||AD| +

|BD||AD|

)=

|AP ||DP | ·

|BC||AD| .

Consequently,

1|BD| +

1|CD| =

|AD| − |DP ||DP | · |AD| =

1|DP | −

1|AD| ,

and1

|DP | =1

|AD| +1

|BD| +1

|CD| . (4)

Conversely, if (3) holds, then

|BC||CD| · |BD| =

|AP ||DP | · |AD| .

so that |CD||AD| =

|BC||BD| ·

|DP ||AP | =

|CE||AE| .

This means that DE bisects ADC. Consequently, DF bisects ADB, and DEF is a right angle.


AMM 10483.959. (S.Rabinowitz)

Given an odd positive integer n, let A1, A2, . . . , An be a regular n−gonwith circumcircle Γ. A circle Oi with radius r is drawn externally tangentto Γ at Ai for i = 1, 2, . . . , n. Let P be any point on Γ between An and A1.A circle C (with any radius) is drawn externally tangent to Γ at P . Let tibe the length of the common external tangent between the circles C and Oi.Prove that

n∑i=1

(−1)iti = 0.

AMM 10514.963.S978. (J.Fukuta)

Let ABC, let P1 and P2, P3 and P4, P5 and P6 be thepoints on the sidesBC, CA, AB respectively, such that

|BP1||P1C| =

|CP2||P2B| =

|CP3||P3A| =

|AP4||P4A| =

|AP5||P5B| =

|BP6||P6A| = r,

with 0 ≤ r ≤ 1.Let A′, B′, C ′ be the points of intersections of P1P4 and P2P5, P3P6 and

P4P1, P5P2 and P6P3, respectively.Let QiPiPi+1, i = 1, . . . , 6 be the equilateral triangles built outwards

on the sides of the hexagon P1P2 · · ·P6. Let RiQi−1Qi+1, i = 1, 2, . . . , 6 bethe equilateral triangles built outwards on the diagaonals of the hexagonQ1Q2 · · ·Q6.

(a) Show that the points Q1, A′ and Q4 lie on R1R4.(b) Show that the diagonals R1R4, R2R5 and R3R6 are concurrent, equal

in length, and that the angle of intersection of two of these is 60.(c) Let Gi be the centroid of the triangle Ri−1RiRi+1, i = 1, . . . , 6. Show

that G1G2 · · ·G6 is a regular hexagon and that its center coincides with thecentroid of the triangle ABC.

AMM 10517.964.S979. (J.Anglesio)

Let ABC be a triangle and let H be its orthocenter and I its incenter.If W is the point such that HW = 4HI an R = 2

√2|HI|, show that none

of the vertices A, B, or C is in the interior of the circle with center W andradius R.


AMM 10533.966. (A.Flores)

On a parallelogram P construct exterior squares on the sides. The centersof these squares form a square QE . On the same parallelogram constructthe interior squares on the sides. The centers of these squares form anothersquare QI .

(a) Show that area (QE) - area (QI) = 2 area (P ).(b) Is there a generalization when P is replaced by an arbitrary convex

quadrilateral?

AMM 10542.967.S982. (J.Anglesio)

Let C be the circumcircle of a trinangle A0B0C0 and I the incircle. It isknown that, for each point A on C, there is a triangle ABC having C forcircumcircle and I for incircle. Show that the locus of the centroid G oftriangle ABC is a circle that is traversed three times by G as A traverses Conce, and determine the center and radius of this circle.

AMM 10547.968. (D.Sachelarie and V.Sachelarie)

In the triangle ABC, let O be the circumcenter, H the orthocenter, and Ithe incenter. Prove that the triangle OHI is isosceles if and only if

a3 + b3 + c3

3abc=R

2r.

AMM 10560.9610.S982. (E.Alkan)

Consider a convex quadrilateral ABCD, and choose points P , Q, R, and Son sides AB, BC, CD, DA respectively, with

|PA||PB| =

|RD||RC| and

|QB||QC| =

|SA||SD| .

LetK denote the area ofABCD, and letKA, KB , KC andKD denote the ar-eas of SAP , PBQ, QCR, RDS respectively. Show thatK4 ≥ 212KAKBKCKD

and determine a necessary and sufficient condition for equality.

AMM 10588.974.S994. (Marcin Mazur)

Let A1A2A3 be a triangle. For i = 1, 2, 3, let Bi be a point on side Ai+1Ai+2,where subscripts are taken modulo 3.


(a) Show that |AiBi+1| + |BiBi+1| = |AiBi+2| + |BiBi+2| for i = 1, 2, 3if and only if Bi is the midpoint of Ai+1Ai+2 for i = 1, 2, 3.

(b) Show that |AiBi+1| + |AiBi+2| = |BiBi+1| + |BiBi+2| for i = 1, 2, 3if and only if Bi is the midpoint of Ai+1Ai+2 for i = 1, 2, 3.

AMM 10602.976.S985. (Dan Sachelarie and Vlad Sachelarie)

In triangle ABC, prove that

HIO ≥ π

2+ arcsin

√2rR,

with equality if and only if HIN = π2 .

AMM 10616.978. (Ernesto Bruno Cossi)

Let K ba a compact, covex set in the plane. For each interior point P ofK and each line through P , let A and B be the two points of on theboundary of K, and let Q be the harmonic conjugate of P with respect to Aand B. If K is an ellipse, then for each P the locus of points Q is a straightline. Is the converse true?

AMM 10631.9710. (Greg Huber)

Given a triangle T , let the intriangle of T be the triangle whose verticesare the points where the circle inscribed in T touches T . Given a triangleT0, form a sequence of triangles T0, T1, T2, . . . , in which each Tn+1 is theintriangle of Tn. Let dn be the distance between the incenters of Tn andTn+1. Find limn→∞

dn+1

dnwhen T0 is not equilateral.

AMM 10637.981.S998. (C.F.Parry)

Suppose triangle ABC has circumcircle Γ, circumcenter O, and orthocenterH. Parallel lines α, β, γ are drawn through the vertices A, B, C respec-tively. Let α′, β′, γ′ be the reflections of α, β, γ in the sides BC, CA, ABrespectively.

(a) Show that α′, β′, γ′ are concurrent if and only if α, β, γ are parallelto the Euler line OH.

(b) Suppose that α′, β′, γ′ are concurrent at the point P . Show that Γbisects OP .


AMM 10644.982.S995. (Mihaly Bencze)

Given an acute triangle ABC with sides of length a, b, c, inradius r, andcircumradius R, prove that

r

2R≤ abc√

2(a2 + b2)(b2 + c2)(c2 + a2).

AMM 10653.983.S00(1)89–91. (Marcin Mazur)

Given an isosceles triangle ABC, prove that there is a unique set ofpoints A1, B1, C1 on sides BC, CA, AB respectively, with the propertythat the quadrilaterals AC1A1B1, BA1B1C1, CB1C1A1 circumscribe cir-cles. Furthermore, prove that the inradius of ABC is twice the inradiusof A1B1C1.

AMM 10659.984.S999. (Jiro Fukuta)

Let D, E, F be points in the interior of sides BC, CA, AB respectively,of triangle ABC such that the incircles of AEF , BFD, and CDE arecongruent, each having radius r. Let ρ, s, K be the inradius, semiperimeter,and area of triangle ABC, and let ρ′, s′, K ′ be the corresponding quantitiesfor DEF .

(a) Prove that ρ′ = ρ− r, s′ = (1 − rρ)s, and K ′ = (1 − r

ρ)2K.(b) Prove that, if r = ρ

2 , then D, E, F are midpoints of the sides oftriangle ABC.

See also Crux 1191.

AMM 10662.985.S999. (J. Konhauser and Stan Wagon)

Find a construction for the center of gravity of the edges of a quadrilateral.

AMM 10673.98(6)559.S00(2)180–181. (Marcin Mazur)

Let C be the circle insrcibed in the triangle A1A2A3, and let Pi ∈ Ai+1Ai+2

(subscripts taken modulo 3) be such that the lines PiAi are concurrent. Letti be the second tangent from Pi to C, the first being Ai+1Ai+2. Prove thatthe points Q1, Q2, Q3 defined by Qi = ti ∩ Pi+1Pi+2 are collinear.


AMM 10678.98(7)666.S00(4)373. (Kimberling and Yff)

Let C be the incircle of triangle A1A2A3. Suppose that whenever i, j, k =1, 2, 3, there is a circle through Aj and Ak meeting C in a single pointB − i. Prove that the lines A1B1, A2B2, A3B3 are concurrent.

AMM 10686.98(8)768.S00(7)656–657. (C.R.Pranesachar)

An equicevian point of a triangle ABC is a point P (not necessarily insidethe triangle) such that the cevians on the lines AP , BP , CP have equallength. Let SBC be an equilateral triangle, and let A be chosen in theinterior of SBC on the altitude dropped from S.

(a) Show that ABC has two equicevian points.(b) Show that the common length of the cevians through either of the

equicevian points is constant, independent of the choice of A.(c) Show that the equicevian points divide the cevian through A in a

constant ratio, independent of the choice of A.(d) Find the locus of the equicevian points as A varies.(e) Let S′ be the reflection of S in the line BC. Show that (a), (b), and

(c) hold if A moves on any ellipse with S and S′ as foci. Find the locus ofthe equicevian points as A varies on the ellipse.

AMM 10693.98(9)859.S00(2)182–184. (Wu Wei Chao)

Let P be an arbitrary point on the side BC of triangle ABC.(a) Let D be the point where the incircle of triangle ABC meets BC,

and let Q and R be the incenters of ABP and ACP respectively. Provethat QDR is a right angle.

(b) Let D′ be the point where the excircle opposite A of ABC meets BC.Let Q′ and R′ be the excenters opposite A of ABP and ACP respectively.Prove that QDR ∼ R′D′Q′.

(c) Prove that the lines BC, QR, Q′R′ are concurrent or parallel.

AMM 10698.98(10)995.S00(7)657–658. (Wu Wei Chao)

Let P be the intersection of the two diagonals of a convex quadrilateralABCD. Let the radii of the circles inscribed in the four triangles APB,BPC, CPD, and DPA be r1, r2, r3 and r4 respectively. Show that ABCDhas an inscribed circle if and only if

1r1

+1r3

=1r2

+1r4.


AMM 10703.98(10)956.S00(3)285. (J.Angelsio)

Given triangle XY Z, let its incenter be I, its centroid C, its circumcenterO, its orthocenter H, the center of its nine-point circle W , its Gergonnepoint G, its Nagel point N . Let S denote the intersection of the line IGwith the Euler, and let T , U , V denote respectively the intersections of theline IG with lines NO, NW , and NH.

(a) Show that C lies one third of the way from H to S (so that SO =HO).

(b) Show that ST : SI : SU : SV = 10 : 15 : 18 : 30.(c) Show that NO : TO = 3 : 1; NW : UW = 5 : 3, and NH : V H. [We

may now infer that NH = 2 ·OI and that these segments are parallel].

AMM 10704.991. (Wiliam G Spohn)

Show that there are infinitely many pairs ((a, b, c), (a′, b′, c′)) of primitivePythagorean triples such that |a− a′|, |b− b′|, and |c− c′| are all equal to 3or 4. Examples include ((12,5,13), (15,8,17)) and ((77,36,85), (80,30,89)).

AMM 10710.99(1)68.S00(6)572–573. (Bogdan Suceava)

Let ABC be an acute triangle with incenter I, and let D, E, and F bethe points where the circle inscribed in ABC touches BC, CA, and ABrespectively. Let M be the intersection of the line[s] through A parallel toBC and DE, and let N be the intersection of the line[s] through A parallelto BC and DF . Let P and Q be the midpoints of DM and DN respectively.Prove that A, E, F , I, P , Q are on the same circle.

AMM 10713.99(2)166.S00(5)464. (Juan - Bosco Romero Marquez)

Given a triangle with angles A ≥ B ≥ C, let a, b, and c be the lengths ofthe corresponding opposite sides, let r be the radius of the inscribed circle,and let R the the radius of the circumscribed circle. Show that A is acuteif and only if R+ r < 1

2(b+ c).

AMM 10717.99(2)167.S00(5)466-167. (M.Mazur)

We say that a tetrahedron is rigid if it is determined by its volume, theareas of its faces, and the radius of its circumscribed sphere. We say that atetrahedron is very rigid if it is determined just by the areas of its faces andthe radius of its circumscribed sphere.


(a) Prove that every tetrahedron with faces of equal area is rigid.(b) Prove that a very rigid tetrahedron with faces of equal area is regular.(c)* Is every tetrahedron rigid?(d)* Is every very tetrahedron regular?

AMM 10719.99(3)264.S00(10)952–954. (Jean Anglesio)

Let A, I, and G be three points in the plane. Let M denote the point 2/3of the way from A to I, and let U and V be the circles of radius |AM | eachof which is tangent to AI at M . Show that when G is outside both U andV , there are precisely two triangles ABC with incenter I and centroid G.Provide a Euclidean construction for them. Show that when G is in theinterior of U or V , there does not exist a triangle ABC with incenter I andcentroid G.Solution: Consider a triangle ABC with incenter I and centroid G. LetD be the midpoint of the side BC. Clearly, GD = 1

2AG. Let X be theorthogonal projection of I on BC. This is the point of tangency of theincircle with side BC. Note that X lies on the circle with diameter ID. LetY be the point on BC, symmetric to X with respect to D. This is the pointof tangency of BC with the excircle on this side. The segment AY passesthrough a point N , called the Nagel point of the triangle, which lies on theline IG, with GN = 2IG.

To construct triangle ABC given a vertex A, the incenter I, and thecentroid G, we therefore proceed in the following steps.

(1) Locate point D which divides the segment AG externally in the ratio3 : −2.

(2) Locate point N which divides the segment IG externally in the ratioIN : NG = 3 : −2. Note that AN and ID are parallel.

(3) Construct the circle C with ID as diameter.(4) Extend the segment ND to N ′ such that ND = DN ′, and construct

a line through N ′ parallel to AN .

The intersections of with C, if any, are the possible locations of the pointX, the point of tangency of the incircle with the side BC. The triangle ABCis then bounded by the line DX and the two tangents from A to the circlewhich has center I and passes through X.

It remains to determine the condition for real intersections of and C.Obviously, a necessary and sufficient condition is that N ′ lies on or insidethe circle C.


See MG 1149.824, and W. Wernick, Triangle constructions with threelocated points, Math. Mag. 55 (1982) 227 – 230. This is also a follow-up ofthis article by Meyers around 1995.

AMM 10734.99(4)470.S00(7)658–659. (Floor van Lamoen)

Let ABC be a triangle with orthocenter H, incenter I, and circumcenterO. Let [P, r] denote the circle with center P and radius r. Show that theradical center of [A,CA+AB], [B,AB+BC], and [C,BC+CA] is the pointobtained by reflecting H through O and then reflecting the result throughI.

This is the same as CMJ 664.995 (J. Fukuta).

AMM 10749.997. (Alain Grigis)

Let ABC be a triangle with a right angle at B and an angle of π6 at A.

Consider a billiard path in the triangle that begins at A, reflects sucessivelyoff side BC at P , off side AB at R, off side AC at S, and then ends at B.

(a) Show that AP , QR, and SB are concurrent at a point X.(b) Show that the angles formed at X measures π

3 .(c) Show that AX = XP + PQ+QX = XR+RS + SX = 2XB.

AMM 10755.998. (Jiro Fukuta)

An arbitrary circle O is drawn through vertices B and D of a convex quadri-lateral ABCD. LetO1 be the circle tangent to lines AB and AD and tangentto O internally at a point of O on the opposite side of line BD from A. LetO2 be the circle tangent to lines CB and CD and tangent to O internallyat a point of O on the opposite side of line BD from C. Let R1 and R2 bethe radii of circles O1 and O2, respectively, and let r1 and r2 be the radiiof the incircles of triangles ABD and CBD respectively. Prove that thequadrilateral ABCD is inscribable in a cicle if and only if r1

R1+ r2

R2= 1.

AMM 10756.998. (Douglas Iannucci)

Prove that

cosπ

7=

16

+√

76

(cos(13

arccos1

2√

7) +

√3 sin(

13

arccos1

2√

7)).


AMM 10759.99(8)778.S00(9)867–868. (Calin Popescu)

In triangle ABC, let ha denote the altitude to the side BC and let ra denotethe exradius relative to the side BC, etc. Prove that

hnar

na + hn

b rnb + hn

c rnc ≤ rn

arnb + rn

b rnc + rn

c rna .

AMM 10763.999. (J. Angelsio)

Let ABC be a triangle; let O be its circumcenter, H its orthocenter, I itsincenter, N its Nagel point, and X, Y , Z its excenters. Let S be defined sothat O is the midpoint of HS, and let T denote the midpoint of SN . It isknown that the orthocenter and the nine-point center of triangle XY Z areI and O, respectively. Prove that

(a) the circumcenter of triangle is T ; and(b) the centroid of triangle XY Z is the centroid of SIN .

AMM 10780.00(1)84.S00(10)957–158. (K. Kedlaya)

Let T be a triangle. Two circles in T are called partners if they are theincircles of two triangles with disjoint interior whose union is T . Everycircle tangent exactly to two sides of T has two partners. Let C1, C2, . . . , C6

be distinct circles such that Ci, Ci+1 are partners for each i ∈ 1, 2, 3, 4, 5, 6.Show that C6 and C1 are partners.

See E3236.

AMM 10783.00(2)176. (Wu Wei Chao)

Let ABCD be a cyclic quadrilateral such that AD is not parallel to BC.Given points E and F on the line CD, let G and H be the circumcenters ofBCE and ADF . Prove that the lines AB, CD, and GH are concurrent orparallel if and only if there is a circle through A, B, E, F .

AMM 10796.00(4)367. (Floor van Lamoen)

Let ABC be a triangle and let the feet of the altitudes dropped from A, B,C be A′, B′, C ′ respectively. Show that the Euler lines of triangles AB′C ′,A′BC ′, and A′B′C concur at a point on the nine-point circle of ABC.

AMM 10804.00(5)462. (A. Sinefakopoulos)

Let ABCD be a convex quadrilateral with an incircle that contacts AB at Eand CD at F . SHow that ABCD has a circumcircle if and only if AE

EB = DFFC .


AMM10810.00(6)566. (J-B. Romero Marquez)

Consider a convex quadrilateral with no parallel sides. On each side ABselect a pint T as follows: Draw lines from A and B parallel to the oppositeside. Let A′ and B′ be the new points where these lines intersect the sidesneighbouring AB. Let T be the point where AB intersects A′B′. Prove thatthe four points selected in this way are the corners of a parallelogram.

AMM 10814.00(6)567. (R.Satnoianu)

Let P be in the interior of triangle ABC. Let r, s, t be the distances fromP to the vertices A, B, C respectively, and let x, y, z be the distances fromP to the sides BC, CA, AB respectively. Prove that

(a) qr + qs + qt + 3 ≥ 2(qx + qy + qz) for any q ≥ 1,(b) qs+t +qt+r +qr+s +6 ≥ q2x +q2y +q2z +2(qx +qy +qz) for any q ge1.

AMM 10830.00(9)863. (Floor van Lamoen)

A triangle is divided by its three medians into 6 smaller triangles. Showthat the circumcenters of these smaller triangles lie on a circle.

See also VIS 274. The center is the point

[10a4 − 13a2(b2 + c2) + 2(2b4 − 5b2c2 + 2c4)].

Check!

AMM 10838.00(10)950. (F.S.Parvanescu)

Let M be any point in the interior of triangle ABC, and let D, E, F bepoints on the perimeter such that AD, BE, CF are concurrent at M . Showthat if the triangles BMD, CME, AMF all have equal areas and equalperimeters, then ABC is equilateral.

AMM 10845.01(1)77. (C. Demeter)

Let E be a point inside the triangle ABC such that ABE = ACE. Let Fand G be the feet of the perpendiculars from E to the internal and externalbisectors, respectively, of angle BAC. Prove that the line FG passes throughthe midpoint of BC.


AMM 10862.01(1)271. (Wu Wei Chao)

Let P be a point in the interior of triangle ABC. Prove that PCB = PAC = PBC if and only if BP ·AC = CP ·AB amd BPC + BAC =180.


Mathematics Magazine

1976 – 1996

MG867.733.S742.(Carlitz)

Let P be a point in the interior of ABC. Let R1, R2, R3 denote the dis-tances of P from the vertices of ABC and let r1, r2, r3 denote the distancesfrom P to the sides of ABC. Show that∑

r1R2R3 ≥ 12r1r2r3;∑

r1R21 ≥ 12r1r2r3;

∑r22r

23R2R3 ≥ 12r21r

22r

23.

In each case there is equality if and only if ABC is equilateral and P isthe center of ABC.

MG880.735.S751.(R.Corry)

Given chords a, b, c in a semicircle. Determine the diameter of the circle.Solution. D3 − (a2 + b2 + c2)D − 2abc = 0.

See also AMME574.435.

MG896.742.(S.B.Maurer)

Prove that there are infinitely many Pythagorean triplets of the form (a, a+1, c).

MG898.742.(R.D.H.Jones)

There are five points associated with every triangle: the orthocenter, thecentroid, the incenter, the circumcenter, and the nine-point center. Provethat if any two of these coincide the triangle is equilateral.

MG900.742.(Klamkin)

A long sheet of rectangular paper ABCD is folded such that D falls on ABproducing a smooth crease EF with E on AD and F on CD when unfolded.Determine the minimal area of triangle EFD by elementary methods.


MG901.743.S753.(Bankoff)

The sides of any triangle are rational or integral only if the ratio of theinradius to the circumradius is rational. Is the converse true ? Answere Theconverse is not true. M.Goldberg suggested that the problem could havebeen stated as follows: If the sides a, b, c have rational ratios, then the ratioof the inradius r to the circumradius R is rational. If K is the area of thetriangle and s its semiperimeter, then R = abc

4K (sic!) and r = Ks . Given the

fact that Rr is rational, it does not follow that the ratios of the sides of the

triangle are always rational.

MG910.744.(L.Carlitz)

Let P be a point in the interior of ABC and let r1, r2, r3 denote thedistances from P to the sides of ABC. Let a, b, c denote the sides and rthe radius of the incircle of ABC. Show that

a

r1+

b

r2+

c

r3≥ 2s

r,

ar21 + br22 + cr23 ≥ 2r2s,(s− a)r2r3 + (s− b)r1r3 + (s− c)r1r2 ≤ r2s,ar21 + br22 + cr23 + (s− a)r2r3 + (s− b)r3r1 + (s− c)r1r2 ≥ 3r2s,

where 2s = a + b + c. In each case there is equality if and only if P is theincenter of ABC.

MG913.744.(Garfunkel)

Triangles A1B1C1 is inscribed in a circle. The medians are drawn and ex-tended to the circle meeting the circle at points A2B2C2. The medians oftriangle A2B2C2 are likewise drawn and extended to the circle to pointsA3B3C3, and so on. Prove that triangle AnBnCn becomes equilateral asn→ ∞ (and very rapidly).

MG916.745.(H.Demir)

Let XY Z be the pedal triangle of a point P with regard to the triangleABC. Then find the trilinear coordinates x, y, z of P such that Y A+AZ =ZB +BX = XC + CY .


MG917.745.(Trigg)

The length of every edge of a regular pentagonal prism is e.(a) When the two pentagonal faces are rotated about parallel diagonals

until two of their edges coincide, two lateral edges vanish and one becomeselongated. The resulting hexahedron has two congruent regular pentagons,two congruent equilateral triangles, and two congruent trapezoids for faces.Eleven of its edges are equal. What is the length of the twelfth edge ?

(b) When the pentagonal faces are otherwise rotated about the paralleldiagonals until two of the vertices coincide, one lateral edge vanishes andtwo are elongated. The resulting heptahedrron has two congruent regularpentagons, two congruent equilateral triangles, two congruent trapezoidsand one rectangle for faces. Twelve of its edges are equal. What are thelengths of the other two edges ?

MG919.745.(Klamkin)

An (n+ 1)−dimensional simplex with vertices O, A1, A2, . . . , An+1 is suchthat the (n + 1) congruent edges OAi are mutually orthogonal. Show thatthe orthogonal projection of O onto the n−dimensional face opposite to itcoincides with the orthocenter of that face. (Thsi generalizes the knownresult for n = 2.

MG920.(Bankoff)

The radius of the nine-point circle is equal to rr1r2r3h1h2h3

.

MG925.751.

(a) (J.G.Baron) Prove that any non-self-intersecting cyclic octagon is suchthat the sum of any four nonadjacent interior angles is 3π.

(b) (T.E.Elsner) An octagon is inscribed in a circle with vertices onany four diameters. Show that each alternate pair of exterior angles iscomplementary.

MG929.752.(Trigg)

Show that there are two octahedra with equilateral triangular faces.


MG936.752.(Garfunkel)

It is known that ha+hb+hc ≤√

3s. Prove or disprove the stronger inequalityta + tb +mc ≤

√3s.

MG945.754.S764.(A.Wayne)

Find the smallest Pythagorean triangle in which a square with integer sidescan be inscribed so that an angle of the square coincides with the rightangle of the triangle. Solution. Consider a right triangle whose sides are(ka, kb, kc), with a, b, c relatively prime. The bisector of the right angle haslength k

√2· ab

a+b . Each side of the square insribed in the triangle has rationallength k · ab

a+b . To make this an integer, we take k = a+ b. The area of thetriangle is 1

2ab(a+ b)2. This is smallest when a = 3, b = 4, a+ b = 7. Thetriangle has sides 21, 28, 35, and the inscribed square has side 12.

MG949.754.S764.(Erdos)

In a circle with center O, OXY is perpendicular to chord AB (as shown,MG(1975)p.239). Prove that DX ≤ CY .

MG955.755.(C.F.White)

For three line segments of unequal lengths a, b, and c drawn on a planefrom a common point, characterize the proper angular positions such thatthe outer end-points of the line segments define the maximum-area triangle.Show how to approximate the exact values of the edges for a = 3, b = 4 andc = 5.

MG959.755.(Carlitz)

Let P be a point in the interior of ABC and let r1, r2, r3 denote thedistances from P to the sides of ABC. Let R denote the circumradius.Show that

√r1 +

√r2 +

√r3 ≤ 3

√R

2.

MG960.755.(A.Wayne)

In a rectangle of dimensions a and b, lines parallel to the sides divide theinterior into ab square unit areas. Through the interior of how many of theseunits squares will a diagonal of the rectangle pass?


MG761.p25.

Niven, A new proof of Routh’s theorem: If the sides BC,CA,AB ofABC are divided at points D,E,F in the ratios 1 : λ, 1 : µ, 1 : ν respec-tively, then the area of the triangle formed by AD,BE,CF is

(λµν − 1)2

(λµ+ µ+ 1)(µν + ν + 1)(νλ+ λ+ 1).

See also CM7.p199,273, Three more proofs of Routh’s theorem.

MG963.761.S771.

Characterize convex quadrilaterals with sides a, b, c, d such that the deter-minant of the cyclic matrix with entries a, b, c, d is zero. Solution. Eithera + c = b + d or a = c, b = d. In the first case, the quadrilateral can becircumscribed about a circle. In the

MG966.761.S773.

A point P lies in the interior of rectangle of sides a and b. (i) Find a, b andP so that all eight distances from P to the four vertices and the four sidesare positive integers.

(ii) Find an example of a square where seven of the distances are integers.(iii) Can all eight distances be a square for a square?See also MG813.p131, MG1147.823. MG865.73?, GuyD19..

MG967.761.S773.

Let ABC be a triangle inscribed in a circle with internal bisectors of B andC meeting the circle again in the points B1 and C1 respectively.

(i) If B = C, then BB1 = CC1.(ii) Characterize ABC for which BB1 = CC1. Solution. A = 60

degrees, or B = C.

MG988.764.S781.

A given equilateral triangle ABC is projected orthogonally from a givenplane P to another plane P ′. Show that the sum of the squares of the sidesof A′B′C ′ is independent of the orientation of ABC in the plane P .


MG998.765.S783.

Characterize all triangle in which the the triangle whose vertices are the feetof the internal angle bisectors is a right angle.

Solution. A′C ′B′ is a right angle if and only if ACB = 120 degrees.

MG1010.773.S795.

Given ABC, points D,E,F are on the lines determined by BC,CA,ABrespectively. Assume AD = BE = CF . The lines AD,BE,CF intersect toform PQR.

(i) Show that PQR is equilateral if and only if ABC is.(ii) Express the area of PQR in terms of that of ABC.

MG1020.773.S791.

For i = 1, 2, 3, let the circle Ci have center (hi, ki) and radius ri. Find adeterminant equation for the circle orthogonal to these three given circles.

MG1023.774.S791.

Call a triangle super-Heronian if it has integral sides and integral area, andthe sides are consecutive integers. Are there infinitely many distinct super-Heronian triangles? Solution. With sides a− 1, a, a + 1, = a

4

√3(a2 − 4).

is an integer if and only if 14

√3(a2 − 4) is an integer. Now, an+1 =

a2n − 2 with a1 = 14 defines a recurrent sequence such that 1

4

√3(a2

n − 4) isalways an integer. These gives distinct, nonsimilar super-Heronian triangles.(13, 14, 15; 84).

MG775.p261

Klamkin, An ellipse inequality.

MG1028.775.S793

Let P1, P2, P3 be arbitrary points in the plane of ABC. Let arbitrary linesperpendicualr to APi, BPi, CPi determine AiBiCi, i = 1, 2, 3. Now, letA0, B0, C0 be the respective centroids of triangles A1A2A3, B1B2B3, C1C2C3.Show that the perpendiculars from A,B,C on the sides of A0B0C0 concur.


MG1039.783.S794

Sum the series∑∞

k=11k2 tan kπ

m tan kπn .

MG1043.783.S795

For two triangles,

s1r1R1

s2r2R2

12 ≥ 3 1√

a1a2+

1√b1b2

+1√c1c2

,

with equality if and only if the triangles are equilateral. Also show that theanalogous three triangle inequality

s1r1R1

s2r2R2

s3r3R3

≥ 9 1√a1a2a3

+1√b1b2b3

+1√c1c2c3

is not valid.

MG1054.785.S801

Construct ABC by straightedge and compass given(i) a,ma, wa.(ii) A,ma, wa.Solution by Howard Eves, with an interesting quotation.

MG1057.785.S802.

Disseect a regular pentagon into six pieces and reassemble the pieces to formthree regular pentagons whose sides are in the ratio 2 : 2 : 1.

MG1076.794.S804.(Klamkin)

Let B be an n-gon inscribed in a regular n-gon A. Show that the vecticesof B divide each side of A in the same ratio and sense if and only if B isregular.

See also CMJ794.146.S811, CMJ815.203.

MG1077.794.S804.

Show that the number of integral-sided right triangles whose ratio of area tosemi-perimeter is pm, where p is a prime and m a positive integer, is m+ 1,if p = 2, and 2m+ 1, if p = 2.

See also MG795.1088.S811, CMJ845.p429, Crux 2012.


MG1088.795.S811.

(a) For each positive m, how many Pythagorean triangles are there whichhave an area equal to m times the perimeter? How many of these areprimitive?

(b)* Can this result be generalized to all triangles with integer sides andarea equal to m times the perimeter?

See also MG1077.794.S804, CMJ845.p429 where (a) appears as a con-jecture. Write m = 2k0pk1

1 · · · pkrr as a product of distinct prime powers.

The number of Pythagorean triangles with area m times perimeters is 2r,among these (k0 + 2)(2k1 + 1) · · · (2kr + 1) are primitive. Steven Kleimanand Klostergaard have written an article to discuss an algorithm for solvingpart (b).

MG1107.805.S821.

Determine the maximum value of

sinA1 sinA2 · · · sinAn

giventanA1 tanA2 · · · tanAn = 1.

MG1119.812.S823.

Let ABC be inscribed in a circle and let point P be the centroid of thetriangle. The line segments AP,BP,CP are extended to meet the circle inpoints D,E,F respectively. Prove that

AP

PD+BP

PE+CP

PF= 3.

MG1120.812.S823.

Let ABC be inscribed in a circle and let P be a point in the interior ofthe triangle. The line segments AP,BP,CP are extended to meet the circlein points D,E,F respectively. Describe all points P for which

AP

PD+BP

PE+CP

PF≤ 3.

See also Crux 643.S8.155.


MG1129.814.S825

Find radii r,R, with r < R, so that m circles of radius r form a closed ringwith each circle externally tangent to a circle of radius R, and n circles ofradius r form a closed ring with each circle internally tangent to the circleof radius R.

MG1132.815.S831.

Let AD,BE,CF be cevians of ABC intersecting at P .(i) Show that if AD bisects angle A and BD · CE = DC · BF , then

ABC is an isosceles triangle.(ii) Show that if AD,BE,CF are bisectors and BP · FP = BF · AP ,

then ABC is a right triangle.

MG1147.823.S833.

If O is a point in the interior of rectangle ABCD and OA = a,OB =b,OC = c, what is OD? Given one such triple, what is the maximum areaof the rectangle?

See also MG966.761.S773, MG813.p131, AMME3208.

MG1149.824.S841

Construct ABC given(i) O,Ma, I.(ii) O,Ha, Ta.(iii) Ma,Ha, I.

MG1151.824.S834

Three points P,Q,R move on curves in the plane. At each instant, thenormal at P to the curve on which P is moving coincides with the bisectorof the angle RPQ. Corresponding conditions hold for the points Q and R.

(i) Show that the perimeter of the triangle PQR is constant.(ii) Find examples other than that of an equilateral triangle whose ver-

tices move around a fixed circle.(iii) Does the result in (i) have a dynamic interpretation in terms of three

heavy masses moving on a smooth horizontal table with a light inextensiblestring looped over them?


MG1155.825.S841

A plane intersects a sphere forming two spherical segments. Let S be oneof these segments and let A be the point of the sphere furthest from thesegment S. Prove that the length of the tangaent from A to a variablesphere inscribed in the segment S is a constant.

MG1156.825.S841

a′(s − a) + b′(s − b) + c′(s − c) ≥ √12′, with equality if and only both

triangles are equilateral.

MG1157.825.S841

The interior surface of a wine glass is a right circular cone. The glass containssome wine and is tilted so that the wine-to-air interface is an ellipse ofeccentricity e and is at right angles to a generator of the cone. Prove thatthe area of the ellipse is e times the area of that part of the curved surfaceof the cone which is in contact with the wine.

MG1161.831.S854

Two equilateral triangles are placed so that their intersection is a hexagon(not necessarily regular). The vertices of the equilateral trianlges are con-nected to form an outer hexagon. Show that if three alternate angles of theouter hexagon are equal, then the triangles have the same center.

See also Crux 745.

MG1168.832.S842

Let P be a variable point on side BC of ABC.

MG1170.833.S843

In ABC, the bisectors of the angles are the segments AP,BQ,CR. Ifa = 4, b = 5, c = 6, find the size of angle QPR.

MG1181.835.S845

The cevians AD,BE,CF of ABC intersect at P . If the areas of BDP,CEPand AFP are equal, then P is the centroid G.


MG1187.842.S852

(A three winged butterfly problem) Let the chord AB of circle O be trisectedat C and D. Let P be any point on the circle other than A and B. Extendthe lines PD and PC to intersect the circle in E and F respectively. Extendthe lines EC and FD to intersect the circle in G and H respectively. LetGH and HE intersect AB in L and M respectively. Prove that AL = BM .

MG1191.843.S853

Let P1, P2, . . . , Pn be the vertices of a regular polygon inscribed in the unitecircle. Let An denote the sum, and Bn the arithmetic mean, of the area ofall triangles P1PjPk for 1 < j < k ≤ n. Evaluate An, and show that An

increases as n→ ∞, and evaluate limn→∞Bn.

MG1197.844.S854.

Characterize the triangles of which the midpoints of the altitudes are collinear.

MG1199.844.

In isosceles ABC with AB = AC, let H be the foot of the altitude fromA, and E the foot of the perpendicular from H to AB, M the midpoint ofEH. Show that AM ⊥ EC. Remark. BCE and HAM are similar.

MG1201.845.S855.

Sum Sn =∑n

k=0 2k tan x2k tan2 x

2k+1 , and find limn→∞ Sn.

MG1206.851.S861

Subdivide the side BC of ABC by points B = P0, P1, . . . , Pn−1, Pn = Cin order. If ri is the inradius of APi−1Pi for i = 1, . . . , n, prove that

r1 + . . .+ rn <12ha ln

s

s− a.

MG1211.852

Find the locus of points under which an ellipse is seen under a constantangle.


MG1224.854.S864

Consider the nonconvex quadrilateral ABCD in the picture. Let I and Jbe chosen so that DI = CF and BJ = CE. Let K and L be the pointswhere the line IJ intersects AB and AD respectively. Show that KJ = IL.

MG1230.855.S865

Let ABC and A′B′C ′ be two similar and similarly oriented triangles in aplane. Let AA′A′′, BB′B′′ and CC ′C ′′ be three triangles lying the plane andsimilar and similarly oriented so ABC. Prove that A′′B′′C ′′ is similar andsimilarly oriented to ABC.

MG1232.861.S871

Let l be the Euler line of the nonisosceles triangle ABC and let d be theinteral angle bisector of C. Prove that

(i) l is perpendicular to d if and only if γ = π3 .

(ii) l is parallel to d if and only if γ = 2π3 .

MG1236.861.S871

Let the functions f and g be defined by

f(x) =π2x

2π2 + 8x2and g(x) =

8x4π + πx2

for all real x. (a) Prove that if A,B,C aare the angles of an acute-angledtriangles, and R its circumradius, then

f(A) + f(B) + f(C) <a+ b+ c

4R< g(A) + g(B) + g(C).

(b)* Determine functions f and g, where f(x) and g(x) have the formx

u+vx2 , with u and v real constants, for which the inequalities above are bestpossible.

MG1238.862

(a) Prove the interior of a triangle contains a point P for which the threetriangles APB,BPC,CPA have congruent incircles.

(b)* Is P uniquely determined? Can the radii be detemined ? What canyou say abolut the property of P?


MG1256.865.S881

Let ABCD be a cyclic quadrilateral, let the angle bisectors at A and B meetat E, and let the line through E parallel to side CD intersect AD at L andBC at M . Prove that LA+MB = LM .

MG1261.871.S882

(a) What is the area of the smallest triangle with integral sides and integralarea?

(b)* What is the volume of the smallest tetrahedron with integral sidesand inte

See also AMME3279.

MG1265.872

Determine the maximum area of a triangle if one sides is of length λ andtwo of its medians intersect at right angles.

MG1278.875.S885

If O is a given point on the prolongation of diameter BA of a given semicir-cle, and if ODC is a secant cutting the semicircle in D and C, prove thatquadrilateral ABCD has maximum area when the orthogonal projection ofDC on AB is equal to the radius of the semicircle.

MG1287.881.S891

Let P be an interior point of the rectangle ABCD. Draw lines throughA,B,C,D perpendicular to PA,PB,PC,PD respectively. Show that thearea of the convex quadrilateral enclosed by these four lines is equal to orgreater than twice the area of the rectangle. When do we have equality?

MG1295.882.S892

Let R be a given rectangle. Construct a square outwards on the length of R;construct another square outwards on the length of the resulting rectangle.Continue this process anticlockwise indefinitely.

(a) Prove that the centers of the spiraling squares lie on two perpendic-ular lines.

(b) As the process continues, show that the ratio of the sides of therectangles approaches the golden mean.


MG1298.883.S893

A quadrilateral ABCD is circumscribed about a circle, and P,Q,R, S arethe points of tangency of sides AB,BC,CD,DA respectively. Let a =AB, b = BC, c = CD, d = DA, p = QS, q = PR. Show that ac

p2 = bdq2 .

MG1305.884

Let P0 = B,P1, P2, . . . , Pn = C be points, taken in that order, on the sideBC of the triangle ABC. If r, ri and h denote respectively the inradii ofABC,APi−1Ai and the common altitude, prove that

n∏i=1

(1 − 2rih

) = 1 − 2rh.

MG1307.884.S895

Let ABC be a triangle with altitudes ha, hb, hc, and P a point inside oron the boundary of the triangle. Show that

PA+ PB + PC ≥ 23(ha + hb + hc)

with equality if and only if the triangle is equilateral and P is its center.

MG1316.891.S901

Characterize the Heronian triangle in which the Eulerian segment OH sub-tends a right angle at the vertex A. (A Heronian triangle is one with integersides and integer area.)

MG1320.892.S902

Let C(I) be a circle with center I, and D,E,F the points of intersectionof C(I) with the lines from I perpendicular to the the sides BC,CA,ABrespectively. Show that AD,BE,CF are concurrent.

MG1322.893.S902

An n-gon of consecutive sides a1, a2, . . . , an is circumsrcibed about a circleof unit radius. Determine the minimum value of the products of all its sides.


MG1237.894.S904

Let the sides PQ,QR,RS.SP of a convex quadrangle PQRS touch an in-scribed circle at A,B,C,D and let the midpoints of the sidesAB,BC,CD,DAbe E,F,G,H. Show that the angle between the diagonals PR,QS is equalto the angle btween the bimedians EG and FH.

MG1333.895.S904

Prove that the quadrilateral formed by the adjacent quadrisectors of theangles of a rhombus is a square.

MG1340.901

Show that the 3 degree angle is the only constructible angle of prime degreemeasure.

MG1351.903

In the acute triangle ABC, let D be the foot of the perpendicualr from Ato BC, let E be the foot of the perpendicular from D to AC and let F bea point on the line segment DE. Prove that AF is perpendicular to BE ifand only if FE/FD = BD/DC.

MG1354.904.S914

Let ABCD be a convex quadrialteral in the plane with trisection pointsjoined as in the figure to form nine smaller quadrilaterals.

(a) Show that the area of A′B′C ′D′ is one-ninth the area of ABCD.(b) Give necessary and sufficient conditions so that all nine quadrilaterals

have equal area.

MG1356.904.S914

Let P,Q be points taken on the side BC of ABC, in the order B,P,Q,C.Let the circumcircles of PAB,QAC intersect at M = A and those ofPAC,QAB at N . Show that A,M,N are collinear if and only if P andQ are symmetric in the midpoint A′ of BC.


MG1362.904.S915

3∑

(∏

a− 1

ni +

∏b− 1

ni +

∏c− 1

ni ) ≤

∏(si

riRi)

1n ≤ 2n

∏(a

− 1n

i + b− 1

ni + c

− 1n

i ),

where the sums and products are over i = 1, . . . , n.

MG1364.911.S921

The incircle of ABC touches BC,CS,AB at points D,E,F respectively.Let P be any point inside triangle ABC. Line PA meets the incircle attwo points; of these let X be the point that is closer to A. In a similarmanner, let Y and Z be the points where PB and PC meet the incirclerespectively. Prove that DX,EY,CZ are concurrent.

MG1371.912=MG1377.913,S922

LetD,E,F be points on the sidesBC.CA,AB of ABC. Let U, V,W,X, Y,Zbe the midpoints of BD,DC,CE,EA,AF,FB respectively. Prove that

UVW + XY Z − 12DEF

is a constant independent of D,E,F .

MG1372.912

For which angles θ, a rational number of degrees, is it the case that tan2 θ+tan2 2θ is rational?

MG1377.913.S923=MG1371.912

Let DEF be a variable triangle inscribed in ABC, and let U,X, V, Y,W,Zbe the midpoints of the line segments BD,DC,CE,EA,AF and FB respec-tively. Show that

UVW + XY Z − 12DEF

is constant.

MG1386.915.S925.(Fukuta)

Let ABC be an acute-angled triangle, let H be the foot of the altitude fromA, and let D,E,Q be the feet of the perpendiculars from an arbitray pointP in the triangle onto AB,AC,AH respectively. Prove that

AB · AD −AC ·AE = BC · PQ.


MG1395.922.S932.(Sadovenau)

Let A1A2 · · ·An be an n−gon circumscribing a circle, and let B1, B2, . . . , B−n denote the points of tangency of thesides. Let M be a point on thecircumference of the incircle. Show that

n∏i=1

d(M,Bσ(i)Bσ(i+1)) =n∏

i=1

d(M,AiAi+1).

MG1396.922.(Fukuta)

Let ABC be an arbitrary triangle, let L1 and L2 be the trisection pointsof BC, arranged in order from B to C. Describe a method for dissectingtriangle ABL1 into four parts, each of which is a triangle or a quadrilateral,so that the parts can be reassembled to form a triangle congruent to triangleAL2C.

MG1402.923.S933.(Pirvanescu)

Let ABC be a given triangle, and M , N , and P be arbitrary points inthe interiors of the line segments BC, CA and AB respectively. Let linesAM , BN and CP intersect the circumcircle of ABC in points Q, R and Srespectively. Prove that

AM

MQ+BN

NR+CP

PS≥ 9.

Same as Crux 1430.

MG1405.924.

Two circles inscribed in distinct angles of a triangle are isogonally relatedif the tangents from the third vertex not coinciding with the sides are sym-metric with respect to the bisector of the third angle. Given three circlesinscried in distinct angles of a triangle, prove that if any two of the threepairs of circles are isogonally related that so is the third pair.

MG1409.925.S935.(G.A.Heuer)

Does there exist a convex pentagon, all of whose vertices are lattice pointsin the plane, with no lattice points in the interior.

See also Putnam Competition A3, 1990.


MG1411.925.S935.(Covas)

Let C1 and C2 be nonconcentric circles in the plane, and consider the set oflines that intersect C1 and C2 in equal chords. Show that these lines are alltangent to a single parabola.

MG1418.932.S942.(Hentzel and Sprague)

Given three distances a, b, c, construct (using straightedge and compass andwith analytic geometry) a square ABCD and a point P such that PA = a,PB = b and PC = c. See E391.397.S406.

MG1420.932.S942.(Turcu)

If α, β, γ, δ areeal numbers, and n is an odd integer, cosα+ cos β + cos γ +cos δ = 0, and sinα+ sin β + sin γ + sin δ = 0, prove that cosnα+ cosnβ +cosnγ + cosnδ = 0, and sinnα+ sinnβ + sinnγ + sinnδ = 0.

MG1421.932.S942.(Fukuta)

If a polygon A1A2 · · ·An has an inscribed circle with center I and a circum-circle with center O, and Ci is the circumcenter of the triangle IAiAi+1,i = 1, 2, . . . , n, where An+1 = An, prove that the C ′

is are concyclic.

MG1423.933.S943.(Mocanu)

Two equilateral triangles, of side lengths a and b respectively, are enclosedin a unit equivlateral triangle so that they have no common interior points.Prove that a+ b ≤ 1.

MG1426.933.S943.(Fukuta)

Consider a circle with center at O, and a regular n−gon A1A2 · · ·An, con-tained entirely with the given circle. Let C denote the center of the n−gon.Let PiQi, i = 1, 2, . . . , n be the chords of the given circle that are perpen-dicular to CAi at Ai. Prove that

∑ni=1(CP

2i + CQ2

i ) is constant.

MG1429.934.S944.(Wee Liang Gan)

Let P be a point inside the convex n−gon A1A2 · · ·An. Prove that at leastone of the angles PAiAi+1, i = 1, 2, . . . , n, is less than or equal to (1

2 − 1n)π.

(All subscripts are taken modulo n).


1431.934.S944.(Fukuta)

In the given triangle ABC, let AD, AE be any cevians from A to BC.Acircle drawn through A cuts AB, AC, AD, AE, or their extensions, at thepoints P , Q, R, S respectively. Prove that

AP ·AB −AR ·ADAS · AE −AQ ·AC =

BD

EC,

where AP , AB, . . . , denote the lengths of the directed line segments AP ,AB, . . . .

MG1435.935.S945.(Pirvanescu)

Let A1, . . . , An be point masses (n ≥ 3) on the spehere S(O,R) of radiusR and cneter O, and let G be their centroid. Let M be an arbitrary pointin thesphere having OG as a diameter, and let Bk be the other intersectionof MAk with the sphere S(O,R). Show that

n∑k=1

MBk ≥n∑

k=1

MAk.

MG1439.941.S951.(C.Vanden Eynden)

All lines in the sketch below have slopes 0, ] ± 1.What point do the pointsPn approach ?

MG1442.941.S951.(W.O.Egerland and C.E.Hansen)

Prove that two ellipses with exactly one focus in common intersect in atmost two points.

MG1444.942.S952. (C.Turcu)

In the following figure, ABCD is a trapezoid with AB parallel to CD, andthe length of AB is the sum of the lengths of AC and CD. E is the midpointof BD, and F is a point on AC such that EF is parallel to CE. Prove that

(a) AE and DF are perpendicular to BF ;(b) C is the incenter of triangle DEF if and only if AD is perpendicular

to AB;(c) EF is parallel to AD if and only if the length of AB is 3 times the

length of CD.


MG1445.942.S952. (I.V.Burkov)

Let r1, r2, r3 and s1, s2, s3 be orthonormal right oriented triplets of vectorsin R3, and k1, k2, k3 be nonzero real numbers with different absolute valuessuch that

k1(r1 × s1) + k2(r2 × s2) + k3(r3 × s3) = 0.

Prove that ri is parallel to si, i = 1, 2, 3.

MG1447.942.S952. (Pirvanescu)

Let M denote an arbitrary point inside or on a tetrahedron A1A2A3A4, andlet Bi be a point on the face Fi opposite vertex Ai, i = 1, 2, 3, 4. For eachi, let Mi be the point where the line through M parallel to AiBi intersectsFi. Show that

min1≤i≤4

≤4∑

i=1

MMi ≤ maxi≤i≤4

AiBi.

MG1452.943.S953. (J.Frohliger and A.Zeuke)

Let ABC be a given triangle and θ an angle between −90 and 90. Let A′,B′, C ′ be points on the perpendicular bisectors of BC, CA, AB respectively,so that BCA′, CAB′, and ABC ′ all have measure θ. Show that for allbut two values of θ, the lines AA′, BB′ and CC ′ are concurrent, providedthat points A′, B′, C ′ ar not equal to A,B,C respectively.

MG1455.944.S954.(Fukuta)

In a hexagon A1A2A3A4A5A6 inscribed in a circle with center O, let Mi,i = 1, 2, . . . , 6 be the midpoints of the sides AiAi+1, where A7 = A1. Provethat if M1M3M5 and M2M4M6 are equilateral, A1A3A5 and A2A4A6

are also equilateral.

MG1457.944.S954. (L.Carter et al.)

(a) For a point P inside a circle draw three chords through P making six60 angles at P and form two regions by coloring the six “pizza slices”alternately black and white. Prove tht the region containing the center hasthe larger area.

(b)* Prove that if five chords make ten 36 angles at P , then the regioncontaining the center has the lesser area.


See also in the same issue, p.267: Proof without words: Fair allocationof a pizza.

MG1460.945.S955.(D.P.Anastssiu)

Let ABC be an acute triangle with altitudes AA′, BB′, CC ′. Let A1,B1, C1 be the second intersection points of lines AA′, BB′, CC ′ with thecircumcircle of triangle ABC. Show that

AA21 sin 2A+BB2

1 sin 2B + CC21 sin 2C > 24S0,

where S0 denotes the area of triangle A′B′C ′.

MG1461.945.S955. (V.Konecny)

Given the vertices V1, V2 and foci at F1, F2 of two parabolas with the sameaxis, construct a common tangent, if one exists, using only a compass andstraightedge. Assume that the unit of length is given.

MG1469.952. (R.Izard)

In EDB shown below, A and C lie on EB and ED respectively; CB andDA intersect at F . Also,

EDBECA = 6,

DC · AB = 4, and CFA + DFB = 145 . Prove that DEB is a right

triangle.

MG1472.952. (E.Grel)

Let Q denote an arbitrary convex quadrilateral inscribed in a fixed circle,and let F(Q) be the set of inscribed convex quadrilaterals whose sides areparallel to those of Q. Prove that the quadrilaterals in F(Q) of maximumarea is the one whose diagonals are perpendicular to one another.

MG1474.953.S963. (G.A.Edgar)

Consider triangle ABC with sides lengths a, b, c. Suppose r, r′, r′′ arepositive numbers satisfying

r′ ≤ a, r′′ ≤ b ≤ r′′ + r, r′ ≤ c ≤ r′ + r;


r′ ≥ 2r, r′′ ≥ 2r, r′′ ≤ 43r′.

What is the least possible measure of angle A?

MG1483.954.S964. (A.Teodorescu-Frumosu)

Let ABC be an arbitrary triangle, and let a, b, c be the lengths of the sidesBC, CA, AB respectively. Let M be the midpoint of the segment BC, letα = BAM , β = CAM , and x = AMB. Show that

b

sinα=

a cos xsin(α− β)

.

MG1487.955.S965. (E.Kitchen)

Given circles C and C′ with centers O and O′, and circles C1 and C2 tangentto C at points M1 and M2, and internally tangent to C′ at points N1 andN2, prove that the lines M1N1, M2N2 and OO′ are concurrent.

MG1493.961.S971. (J.Fukuta)

In ABC, let L1 and L2, M1 and M2, N1 and N2 be distinct points on thesides BC, CA, AB, respectively, such that

BL1

L1C=CL2

L2B=CM1

M1A=AM2

M2C=AN1

N1B=BN2

N2A< 1.

Let PL1L2, QM1M2, RN1N2, SM2N1, TN2L1, and UL2M1 be the equilat-eral triangles built outwards on the sides of the hexagon L1L2M1M2N1N2.

(i) Prove that the segments PS, QT , and RU have equal lengths, andthe lines PS, QT , RU interesect at 120 and are concurrent.

(ii) If G1, G2, G3, G4, G5, G6 are the centroids of triangles QSR, SRT ,RTP , TPU , PUQ, UQS, prove that G1G2G3G4G5G6 is a regular hexagonwhose centroid coincides with that of ABC.

MG1500.963. (S.Stahl)

Let r be a positive real number and let A0B0C0 be equilateral. For eachn ≥ 0, let An+1 and Bn+1 divide the sides AnBn and AnCn respectively inthe internal ratio r : 1, and set Cn+1 = An. If P = limn→∞AnBnCn,prove that the measures of B0PC0, C0PA0, and A0PB0 form an arith-metic progression.


MG1506.964. (WU Wei Chao)

Let I and O denote the incenter and circumcenter, respectively, of ABC.Assume ABC not equilateral. Prove that AIO ≤ 90 if and only if2BC ≤ AB + CA, with equality holding only simultaneously.

MG1649.022. (K. R. S. Sastry)

Prove that if a right triangle has all sides of integer length, then it has atmost one angle bisector of integral length.

The bisector of the right angle has irrational length aba+b sin 45. If both

of the other two bisectors have rational lengths, then sin A2 , cos A

2 , sin B2 ,

and cos B2 are all rational. This would have made sin 45 = sin

(A2 + B

2

)=

sin A2 cos B

2 + sin B2 cos A

2 rational, a contradiction.


College Mathematics Journal

1976 – 1997

CMJ33.751.S761.

Let P be any point on a circle. Prove that the four distances from P to thevertices of a square inscribed in the circle cannot be all rational.

CMJ47.754.S764.

sin2A+sin2B+sin2 C = 2(sinA sinB cosC+sinA cosB sinC+cosA sinB sinC).

CMJ49.754.S764.


sinA+ sinB + sinCcot A

2 + cot B2 + cot C

2

.

CMJ762.p59.

A simple proof of the reflection property of parabolas.

CMJ61.762.S773.

Construct the altitude to the hypotenuse of right triangle T0. Call one ofthe two subtriangles T1. Construct the altitude to the hypotenuse of T1 andcall one of the subtriangle T2. Continue the process so that, in general, Tn

is one of the two subtriangles formed by constructing the altitude to thehypotenuse of Tn−1. In the solution to CMJ13.744, it was shown that thereexist sequences Ti, i = 1, 2, 3, . . . for which

∑∞i=1 AreaTi equals the area of

T0. In any one of these sequences, let hi be the altitude to Ti−1, i = 1, 2, , . . ..Prove that

∑∞i=1 hi is twice the area of T0 divided by the difference between

the hypotenuse and a leg of T0.

CMJ63.762.S774.

Find all values of k for which∑∞

n=1 tank 1n converges.

Answer : k > 1.


CMJ64.762.S774.

An integer sided square is inscribed in an integer-sided right triangle so thata side of the square lies on the hypotenuse. What is the smallest possiblelength of the side of the square? Solution. The length of each side of thesquare is abc

ab+c2 . Is it clear that this has the same answer as CMJ390.885.S902? See also MG945.754.S764.

CMJ72.763.S781.

In angleABC, a > b > c. Prove or disprove

(a− b

c+b− c

a+c− a

b)(

c

a− b+

a

b− c+

b

c− a) = 1 − 8 sin

A

2sin

B

2sin

C

2.

This is true.

CMJ74.764.S781.

Let H be the intersection of the altitudes of acute triangle ABC. ChooseB′ on HB and C ′ on HC so that AB′C and AC ′B are right angles. Provethat AB′ = AC ′.

CMJ75.764.S781.

Find an integer sided triangle such that each of its angles can be trisectedwith straightedge and compasses. Solution. Let S be the set of all cubes ofcomplex numbers a + bi with a2 + b2 ∈ Z. Writing c + di = (a + bi)3, weobtain a right triangle with legs c and d.

CMJ79.771.S782.

In ABC, let D,E,F be points on BC,CA,AB respectively such thatAF ·AB = BD ·BC = CE ·CA = r. Prove that the ratio of the area of thetriangle determined by AD,BE,CF to the area of ABC is 4 − 3

r2−r+1 .

CMJ85.772.S783.

(a) 27Rr ≤ 2s2.(b) Let O be a point within ABC and d1, d2, d3 be the distances from

O to the sides BC,CA,AB respectively. Prove that

d1 sinA+ d2 sinB + d3 sinC =abc

4R2.


CMJ773.p142.

Some consequences of a property of the centroid of a triangle: For any pointP , not necessarily in the plane of ABC,

PA2 + PB2 + PC2 = GA2 +GB2 +GC2 + 3PG2.

CMJ773.p152.

Quasi Pythagorean triples for obligue triangles.

CMJ94.773.S785.

Given a, b+ c and A, 0 < A < π, prove that there exists ABC if and onlyif a < b+ c ≤ a

sin A2

.

CMJ98.774.

Prove or disprove(i). a3 + b3 + c3 ≥ 8

√3s3 . (ii). a4 + b4 + c4 ≥ 162.

CMJ781.p21.

Polya, Guessing and Proving.

CMJ107.781.

Prove or disprove that the radius of a circle inscribed in a Pythagoreantriangle is an integral multiple of the greatest common divisor of the threesides.

CMJ109.781.

If A = π3 , then

=4R2(sin2A+ sinB sinC) − b2 − c2

2 cscA− 4 cotA.


CMJ110.781.

In ABC, AP,BQ,CR are the altitudes, AD,BE,CF the internal bisec-tors of the angles. Let BE,CF intersect AP at X1,X2 respectively, CF,ADintersect BQ at Y1, Y2 respectively, and AD,BE intersect CR at Z1, Z2 re-spectively. Prove that

IX1 · IY1 · IZ1 = IX2 · IY2 · IZ2 = X1X2 · Y1Y2 · Z1Z2.

CMJ117.783.

Let E be the intersection of the diagonals of parallelogram ABCD and letP and Q be points on a circle with center E. Prove that

PA2 + PB2 + PC2 + PD2 = QA2 +QB2 +QC2 +QD2.

CMJ118.783.

Triangles ABC and DEF are inscribed in the same circle. Prove that

sinA+ sinB + sinC = sinD + sinE + sinF

if and only if the perimeters of the triangles are equal.

CMJ120.783.

Let sinA+cosB = p and cosA+sinB = q, where p and q are not both zeroand p2 + q2 ≤ 4. Express sin(A+B), cos(A+B), sin(A−B) and cos(A−B)in terms of p and q.

Three of these, except cos(A+B) are very easy to find.

CMJ125.784.S802.() = CMJ195.813.S825.()

Determine all values of x ∈ (0, π) which satisfy

tan x = tan 2x tan 3x tan 4x.

Answer. kπ18 , k = 1, 5, 7, 11, 13, 17.

CMJ128.784.S802.

tanπ

14(cos

π

14+ cos

3π14

+ cos5π14

) =12.


CMJ130.784.S802.

Prove that(i) 1

a + 1b + 1

c ≥ 92s .

(ii) a2 + b2 + c2 ≥ 4s2

3 .(iii) s2 ≥ 3

√3.

(iv) a2 + b2 + c2 ≥ 4√

3.(v) a3 + b3 + c3 ≥ 8s3

9 .Interesting editorial comments.

CMJ131.791.S803.

The lengths of the sides of an isosceles triangle are integers and its area isthe product of the perimeter and a prime. What are the possible values ofthe prime? Answer. 2, 3, 5.

CMJ134.791.S803.

Let A be the surface area of a rectangular parallelepiped, V the volume andd the diagonal. Prove that 2d2 ≥ A ≥ 6v

23 .

CMJ140.792.S804.

Locate a point P in the interior of a triangle such that the sum of the squaresof the distances from P to the sides of the triangle is a minimum.

CMJ143.793.S805.

Let AD,BE,CF be the medians of ABC. Prove that cotDAB+cotEBC+cotFCA = 3(cotA+ cotB + cotC).

CMJ146.794.S811.

Prove that the smallest regular n-gon which can be inscribed in a givenregular n-gon will have its vertices at the midpoints of the sides of the givenn-gon.

cf. MG1076.794.S804, CMJ203.815.


CMJ153.794.S812.

CMJ156.801.S813.

CMJ159.801.S813.

arctan1

n+ 1+

n∑i=1

arctan1

i2 + i+ 1=π

4.

CMJ160.801.S813.

Triangles ABC and A′B′C ′ are inscribed in a circle so that c− c′ = a− a′.Prove that

sinA+ sinB + sinC = sinA′ + sinB′ + sinC ′

if and only if

(a− a′)3 + (b− b′)3 + (c− c′)3 = 3(a− a′)(b− b′)(c− c′).

CMJ162.802.S814.

What is the minimum number of acute angle triangles into which a squarecan be partitioned ?

Note: This appears in CM29.S1(45).

CMJ165.802.S815.

CMJ175.804.S821.

A triangle is equilateral if and only if = 16(hab+ hbc+ hca).

CMJ187.812.S824.

Find the length of a side of an equilateral triangle in which the distancesfrom its vertices to an interior point are 5, 7, 8.

Long solution by H.Eves. See also CM10.p242, Bottema, On the dis-tances of a point to the vertices of a triangle. Here is a generalization:

For three positive numbers a, b, c satisfying a ≤ b ≤ c ≤ a+b, givea euclidean construction of an equilateral triangle ABC togetherwith a point P (not necessarily inside the triangle) such that AP =a,BP = b and CP = c. Distinguish between the cases a+ b = c anda+ b < c.


Solution. (a) If a + b = c, let O,A,C,B be points on the plane suchthat OA = a,OC = a + b = c,OB = b, AOC = COB = 60 and AOB = 120. Then ABC is an equilateral triangle.

(b) Suppose a + b < c. Let XY Z be a triangle with Y Z =√

3a,ZX =√3b and XY =

√3c. Let A,B and C be the centers of the equilateral tri-

angles with sides Y Z,ZX and XY respectively drawn outside the triangle.It is well known that ABC is an equilateral triangle. The circumcircle ofthe equilateral triangles with centers A,B,C intersect at a point P . Clearly,AP = AX = a, and similarly, BP = b, CP = c. A second equilateral trian-gle A′B′C ′ and a second point P ′ is obtained by constructing the equilateraltriangles “inside” XY Z.

CMJ812.p149.

M.K.Siu, From an inequality to inversion.

CMJ813.p206.

A visual application of the compound angle formula in terms of areas.

CMJ193.813.S825

AE is a chord bisecting angle A of ABC. A circle that is tangentialinternally to the sides AB and AC is also tangential at E to the circumcircleof the triangle. Let H be the center and r0 the radius of this inner circle. IfI is the incenter and r the inradius of the triangle, prove or disprove that

(i) r0 = r sec2 A2 ;

(ii) IH = 2R sin2 A2

1+sin A2

.

Remark. Durell,p.275, sin2A+ sin2B + sin2 C = 2 + 2 cosA cosB cosC.

CMJ195.813.S825 = CMJ125.784.S802.

Find an acute angle with measure x such that

tan x = tan 2x tan 3x tan 4x.

CMJ814.p271.

Vector identities from quaternions.


CMJ196.814.S831.

Assume that l is a line segment which is not a median and which bisects atriangle into two polygons of equal areas. Can l contain the centroid?

CMJ198.814.S831.

CMJ203.815.

Find the largest and the smallest regular polygons of n sides which can beinscribed in a given regular polygon of (n+ 1) sides.

See also CMJ146.794.S811, MG1076.794.S804.

CMJ207.821.S832.

Evaluate[ n2]∑

k=1

cos2 (2k − 1)π2n

.

Answer. n4 .

CMJ821.p61.

A classroom approach to Pythagorean triples.

CMJ218.823.S834.

Prove or disprove that

tan3π11

+ 4 sin2π11

=√

11.

This is true.

CMJ219.823.S841.

Show that the product formula

2π

= cosπ

4· cos π

8· cos π

16· · ·

generates successive approximations of π with error bound En < 13·22n−4 ,

where n is the power of two occurring in the last cosine term used.


CMJ225.824.S842.

Prove or disprove(a) sin2A + sin2B + sin2 C = 2(sinA sinB cosC + sinA cosB sinC +

cosA sinB sinC) True.(b) sin2A+ sin2B + sin2 C < 2(sinA sinB + sinB sinC + sinC sinA)

False.(c) 8(cos A

2 + cos B2 + cos C

2 ) ≥ 27(tan A2 + tan B

2 + tan C2 ). False.

(d) 9√

32 (a2 + b2 + c2) ≥ (a+ b+ c)2(sinA+sinB+sinC) ≥ 54. True.

CMJ227.824.S842.

Let ABCD be a convex quadrilateral with AC and BD intersecting at M .Assume that P and Q are points such that MP ⊥ AB and PM intersectsCD at Q. Prove or disprove

(i) AB2 + CD2 = AD2 +BC2 if and only if AC ⊥ BD.(ii) If AC ⊥ BD, then ABCD is cyclic if and only if DQ = QC.

CMJ825.p329.

vector approach to the Euler line.

CMJ232.825.S843.

A perfect triangle is defined to have integral sides and area such that theperimeter equals the area. It is known that there are only five perfect tri-angles:

T1 = (6, 8, 10);T2 = (5, 12, 13);T3 = (9, 10, 17);T4 = (7, 15, 20);T5 = (6, 25, 29).

Notice that each of the pairs (T1, T3) and (T1, T5) has a common side. Provethat these pairs can be placed along their common sides to form a largertriangle in each case.

See also CMJ845.p429.

CJM233.825.S843.

(a+ b− c)a(b+ c− a)b(a+ c− b)c ≤ aabbcc.

Equality holds if and only if the triangle is equilateral.


CMJ234.825.S844.

Let E be a point in the plane of parallelogram ABCD and Let K(XY Z)denote the area of XY Z. Prove or disprove that

|K(ADE) −K(ABE)| = K(ACE).

This is indeed true.

CMJ831.p2,832.p154.

Honsberger, the Butterfly Problem and other delicacies from the noble artof euclidean geometry.

CMJ831.p72.

The Steiner-Lehmus theorem as a challenge problem.

CMJ238.831.S844.

Let a and b denote the lengths of the legs of a right triangle and c denotethe length of the hypotenuse. Prove that

a2(b+ c) + b2(a+ c)abc

> π.

More generally,a2(b+ c) + b2(a+ c)

abc≥ 2 + csc

C

2.

CMJ831.240.S845.

A circle is externally tangent to the circumcircle of ABC and also to ABat P and AC at Q. Prove that the midpoint of PQ is the center of theescribed circle opposite to A of ABC.

CMJ244.832.S851.

Prove thatn−1∑k=1

csc2 kπ

ncot2

kπ

n=

(n2 − 1)(n2 − 4)45

.


CMJ249.833.S852.

Prove that a point D on BC of ABC exists so that AD is the geometricmean between BD and DC if and only if AC + AB ≤ √

2BC. When doesequality hold ?

Equality holds if and only if AD is the internal bisector of angle BAC.

CMJ834.p301/

Construction of integral cevians. See also CM6.p98, Equal cevians, andpostscript on CM6.p239.

CMJ251.833.S852.

Prove that(a4 + b4 + c4)2 > 2(a8 + b8 + c8)

if and only if a2, b2, c2 are the lengths of the sides of a triangle.

CMJ253.833.S853.

3(cotA+ cotB + cotC) ≥ cotA

2+ cot

B

2+ cot

C

2with equality if and only if the triangle is equilateral.

CMJ262.835.S854.

Assume that a plane π contains both a vertex of a tetrahedron and thecentroid. Prove or disprove that if π separates the tetrahedron into twosolids of equal volume, then π bisects an edge of the tetrahedron.

True.

CMJ835.p382.

Radii of the incircle and excircles of right triangles.

CMJ835.p436.

Ellipses from a circular and spherical point of view.

CMJ264.835.S854.

(2s)2 > 400π61 .


CJM841.p52.

An analytic approach to the Euler line.

CJM841.p55.

Integer sided triangles with one angle twice another. Solution. Easy. Taketwo integers m,n satisfying m < n < 2m and form a = m2, b = n2−m2, c =mn. It is easy to show that 2A = C.

CJM265.841.S854.

Determine the extreme values of

Si = siniB + sini C − siniA

where A,B,C are the angles of a triangle.

CJM268.841.S855.

Show thatn∑

j=1

sec jx · sec(j − 1)x =tan nxtanx

.

CMJ842.p140.

Reflection property of the ellipse and the hyperbola.

CMJ273.842.S861.

Let ABCD be a tetrahedron and let α, β, γ, and δ be the areas of the facesopposite vectices A,B,C,D respectively. If the edges meeting at D aremutually perpendicular, prove that

δ2 = α2 + β2 + γ2.

Interesting solutions. Generalization: The square of the area of oneface of a tetrahedron is equal to the sum of the squares of the areas of theother three faces minus twice the sum of the products of the areas of theother faces two at a time and the cosine of the diheral angle btween them.Proof by vector algebra.


CMJ843.p252.

Proving Heron’s formula tangentially. The traditonal proof is much morebeautiful. Also, CMJ872.p137 Heron’s area formula. This is the standardproof!

CMJ279.843.S862.

Prove or disprove: a concyclic trapezoid is isosceles if and only if the altitudeof the trapezoid is the geometric mean of the bases.

CMJ844.p326.

M.K.Siu On the sphere and cylinder.

CMJ842.p126.

Grazing goat in n dimensions.

CMJ845.p.430.

Return of the grazing goat in n dimensions.

CMJ285.844.S863.

Consider ABC and its excircle opposite A. Secant APRQ meets the sideBC at P and the excircle at R and Q with R between P and Q. Prove ordisprove that AQ

AP is maximum if and only if P is on the inscribed circle ofABC.

CMJ845.p429.

Right triangles with perimeter and area equal.See also CMJ232.825, where it is asserted that there are only five trian-

gles of integer sides, with area equal to perimeter. Two of these are righttriangles, namely (6,8,10) and (5,12,13). This note shows that these are theonly ones that contain right angles.

Conjecture: For every natural number n, there is at least one primitivePythagorean triangle in which the area equals n times the perimeter. SeeMG1088.795.S811, MG1077.794.S804, where this conjecture was resolved.


CMJ286.845.S863.

Prove that ABC is equilateral if and only if

(cotA+ cotB + cotC) · 3√

sinA sinB sinC =32.

CMJ852.p122.

constructing the foci and directrices of a given ellipse.

CMJ300.852.S871.

Let AD be the bisector of angle BAC of ABC. Prove that AD · BC =AC ·AB if and only if angle A is twice one of the other angles of the triangle.

CMJ302.853.S872.

Prove or disprove that a necessary and sufficient condition for

s′(s− a)(s− b)(s′ − c′) > · ′

is C < C ′.Remark. Neuberg-Pedoe Inequality (CM10.p68):

(a2 + b2 + c2)(a′2 + b′2 + c′2) − 2(a2a′2 + b2b′2 + c2c′2) ≥ 16′.

CMJ304.853.S872.

Let ABCD be a convex quadrilateral with consecutive sides of lengthsa, b, c, d. Prove that a necessary and sufficient condition that a circle can beinscribed in ABCD is that

a(a− b+ c+ d)a+ c

=(a+ b− c+ d)d

b+ d.

See also CMJ374.882.S895.

CMJ309.854.S873.

Call a point P in the plane of a triangle ABC a tangency point if thereexist points A′, B′, C ′ for which the following six perpendicular distancesare equal: B to PA′ and PC ′, A to PB′ and PC ′, C to PA′ and PB′.Does every triangle have a tangency point? Solution. If we do not require


that B and C be on the same side of PA′, and similarly for A,C and PB′;A,B and PC ′, then each point on one of the three lines connecting themidpoints of two sides of ABC is a tangency point. But, if we do requirethis condition, then the incenter and the three excenters of the triangle arethe only tangency point.

CMJ317.861.

notation not very inspiring!

CMJ862.p167.

Angling for Pythagorean triples. An acute angle 2θ is Pythagorean if andonly if tan θ is rational.

CMJ325.862.

Let AB be the longest side of a convex quadrilateral ABCD incsrcibed ina circle O. Let OE and OF be the two radii intersecting AB which arerespectively perpendicular to the diagonals DB and AC of ABCD. Let E′

and F ′ be the feet of the perpendiculars to AB from E and F respectively.Prove that E′F ′ is the arithmetic mean of AD and BC.

CMJ863p.238.

Three ways to maximize the area of an inscribed quadrilateral. Among allquadrilateral inscribed in a circle, the square has the largest area. (LeroyMeyer)

CMJ331.864.S883.

notations too complicated inequalities involving sides and angles of a trian-gle.

CMJ334.864.S884.

Let P1P2P3 be a right triangle with right angle at P3. Choose P4 on P1P2

such that P3P4 ⊥ P1P2. Then choose P5 on P2P3 such that P4P5 ⊥ P2P3.Continue this process so that Pi will be a point of Pi−3Pi−2 with Pi−1Pi ⊥Pi−3Pi−2, i = 4, 5, . . .. Provide a euclidean construction for the limit pointof Pi.


CMJ865.p392.

Geometry of the rational plane, not very interesting!

CMJ865.p418.

A pretrigonometry proof of the reflection property of the ellipse.

CMJ865.339.S891.

Suppose ABC is any triangle and D,E,F are arbitrary interior points ofsegments BC,AC,AB respectively. Let G be the intersection of AD andEF , H that of BE and DF , and I that of CF and DE. Prove that

FG

GE· EIID

· DHHF

= (AF

FB· BDDC

· CEEA

)2.

CMJ872.p141.

arctan 12 + arctan 1

3 = π4 without words.

CM292. Fold a square piece of paper to form 4 creases that determineangles of tangents 1, 2, 3.

CMJ346.872.S892.

Given ABC, determine the point P in it (or on the boundary) whichmaximizes the sum PA + PB + PC. (The corresponding problem withmaximum replaced by minimum is well known).

The same problem also appears as Crux 2215.972.

CMJ347.872.S892.

Suppose the cevians AD,BE,CF of ABC mutually trisect each other.Evaluate AF

FB · BDDC · CE

EA .

CMJ351.873.S893.

Let a ≤ b < c be the lengths of the sides of a right triangle. Prove thatc3 + (a+ b)(ab+ c2) > 6abc.

CMJ353.873.S893.

Prove that the interior of a triangle contains a point that lies on threecongruent circles, each tangent to two different sides of the triangle.


CMJ354.873.S893.

Let ABC be an integer sided right triangle. Let CP and CQ be, respectively,the median and altitude of the hypotenuse AB. Under what condition arethe sides of CPQ integers?

CJM874.p300.

Equiangular lattice polygons and semiregular lattice polyhedra. This con-tains the theorem that a regular n-gon can be embedded in a three-dimensionallattice if and only if n = 3, 4, 6. See also Klamkin and Chrestenson, Poly-gons imbedded in a lattice, AMM70 (1963) 447-448. But the main theoremof the present note states that of the thirteen semiregular polyhedra, only thetruncated tetrahedron, the truncated octahedron and the cuboctahedron occuras lattice polyhedra. See also CMJ821.p36, Semi-regular lattice polygons byHonsberger, and also AMM923 by Beeson.

CMJ364.875.S894.

Let P be a point from which exactly two normals can be drawn to the graphof y = x2. Determine the locus of P .

CMJ365.875.S894.

Let A1A2A3A4 be a quadrilateral inscribed in a circle. If Pi is the incenterof AiAi+1Ai+2, i = 1, 2, 3, 4, A5 = A1, A6 = A2, prove that P1P2P3P4 is arectangle.

CMJ366.881.S894.

12rR

≤ 13(1a

+1b

+1c)2 ≤ 1

a2+

1b2

+1c2

≤ 14r2

.

CMJ374.882.S895.

Determine necessary and sufficient conditions on the consecutive sides a, b, c, dof a convex quadrilateral such that one of its configurations has an incircle,a circumcircle and perpendicular diagonals. Solution. See also CMJ304.853.A necessary and sufficient condition for the quadrilateral to have an incircleis that a + c = b + d. A necessary and sufficient condition for the quadri-lateral to have a circumcircle and the diagonals of the quadrilateral to beperpendicular is that a2 + c2 = b2 + d2. Together, these imply a = b, c = dor a = d, c = b. This is the desired condition.


CMJ376.883.S895.

In ABC, AP,BQ,CR are the bisectors, and P ′, Q′, R′ are points onBC,CA,AB such that PP ′//AB,QQ′//BC,RR′//CA. Prove that

1PP ′ +

1QQ′ +

1RR′ = 2(

1a

+1b

+1c).

CMJ382.884.

Let αi, i = 1, 2, 3, 4 be the angles of an inscribed quadrilateral. Prove ordisprove

4∑i=1

√cot

αi

2< 2

4∑i=1

√csc

αi

2.

CMJ383.884.

a2 + b2 + c2 − 2bc− 2ca− 2ab+18abc

a+ b+ c≥ 4

√3.

CMJ385.884.

cosπ

14cos

3π14

cos5π14

=√

78.

CMJ390.885.902.

Find the smallest Pythagorean triangle in which a square with integer sidescan be inscribed so that a side of the square coincides with the hypotenuseof the triangle.

In what sense ? See also CMJ64.762.S774, MG945.754.S764.

CMJ395.891.S903.

If the altitudes of an acute triangle ABC are extended to intersect its circum-circle in points A′, B′, C” respectively, prove that AreaABC ≤ AreaA′B′C ′.

CMJ892.p134.

To view an ellipse in perspective.

CMJ399.892.S903.

Evaluate∑∞

n=1 arctan 2n2 .


CMJ893.p232.

Lattices of trigonometric identities.

CMJ408.894.408.S905.

Let A′, B′, C ′ be the excenters of ABC. The perapendiculars from B′ toAB and from C ′ to AC meet in a point A′′; points B′′ and C ′′ are determinedanalogously. Prove that the lines AA′′, BB′′, CC ′′ are concurrent.

CMJ417.901.S911.

Let ABC be a triangle, the lengths of whose sides are a, b, c. Let I denotethe incenter and E the excenter opposite the vertex A. Let P and Q (S andR) be points on the sides (extensions of the sides) AB and AC such thatsegment PQ (SR) is parallel to side BC.

(a) Prove that (i) PQ = PB +QC if and only if PQ passes through I,and (ii) SR = SB +RC if and only if SR passes through E.

(b) Determine the lengths of the sides of the trapezoid PQRS in termsof a, b and c.

CMJ421.902.S912.

Let P be any point on the median to side BC of ABC. Extend the linesegment BP (CP ) to meet AC(AB) at D(E). If the circles inscribed inAPE and CPD have the same radius, prove that AB = AC.

CMJ903.426.S913.

A circle with center O passes through vectex A of ABC and cuts sidesAB and AC in points D and E respectively. AO extended meets BC at F .If angle AFC is 60 degrees, show that

AD ·AB −AE ·AC = AO · BC.

CJM430.903.S913.

Let ABC be an isosceles triangle with AB = AC. Assume there existspoints Pi, 1 ≤ i ≤ k such that

(a) Pi ∈ AB if i is odd,(b) Pi ∈ AC if i is even,(c) Pi ∈ APi+2if 1 ≤ i ≤ i+ 2 ≤ k,


(d) Pk−1Pk = BC (f k is even Pk = C; if k is odd Pk = B),(e) AP1 = PiPi+1, 1 ≤ i ≤ k − 1.

CMJ443.911.S921.

Let n be a positive integer. Evaluate

1n

n∑k=1

13 + 2 cos 2kπ

n

.

The published solution is a primitive root approach. But I think it iseasier to handle with the theory of equations, starting with the Chebyshevpolynomials.

CMJ444.911.S921.

Given two right or obtuse triangles with sides of lengths a > b ≥ c and A >B ≥ C and perimeters p and P respectively; prove that aA+aP +Ap ≥ pP .

CMJ450.912.S922.

A spherical ball is placed in a corner of a rectangular room so that the ballis tangent to the ceiling and to both walls. What must the ratio of theradius of that ball to the radius of another spherical ball be if the secondball is to be placed between the first ball and the corner and is to be tangentto the first ball as well as the ceiling and the two walls ? Generalize to n-dimensional hyperspheres. Solution. Let R and r be the radii of the spheres.n(R− r)2 = (R+ r)2. From this r

R =√

n−1√n+1

.

CMJ455.913.S923.(R.S.Tiberio)

Characterize those triangles such that the angle bisectors from one vertex,the median from a second vertex, and the altitude from the third vertex areconcurrent.

Solution. cosA = cb+c .

See also AMME263.37p104,599 concerning the euclidean construction ofthis triangle.

CMJ923.p106

is a very interesting reprint of Bankoff on the shoemaker’s knife. See alsoCM3.p60,p.216.


CMJ923.p118, Johnson’s theorem:

If three circles of equal radii intersect in a point O, their remaining inter-section points lie on a circle of the same radius. Moreover, the orthocenterof these three points is O.

CMJ462.915.S925.(J.Fukuta)

Through any point O inside a given equilateral triangle ABC, lines UR, QTamd SP are drawn parallel to the sides BC,CA,AB respectively, where P ,Q; R, S; T , U are on the sides BC, CA and AB respectively. PRove thatthe area and perimeter of the hexagon PQRSTU are both minimized whenO is the centroid of the triangle.

CMJ467.921.(Rabinowitz)

The cosines of the angles of a triangle are in the ratio 2 : 9 : 12. Find theratio of the sides of the triangle.

See also CM10.p36, Bottema and Sauve. Solution. Let A,B,C be thevertices of the triangle, H its orthocenter, and X,Y,Z the projections of Hon the sides BC,CA,AB respectively. If a, b, c denote the lengths of thesides of the triangle, and α, β, γ their opposite angles, it is well known that

AH : BH : CH = cosα : cos β : cos γ.

From this it follows that

1a

:1b

:1c

= AX : BY : CZ= AH +HX : BH +HY : CH +HZ= cosα+ cos β cos γ : cos β + cos γ cosα : cos γ + cosα cos β.

It remains to determine the cosines of the angles. Since cosα : cos β :cos γ = 2 : 9 : 12, we may write

cosα = 2x, cos β = 9x, cos γ = 12x

for some positive number x. Since

2 cosα cos β cos γ + cos2 α+ cos2 β + cos2 γ − 1 = 0,

we have432x3 + 229x2 − 1 = 0.


It is easy to check that x = 116 is the only positive root of this equation.

From this,

cosα =18, cos β =

916, cos γ =

34.

Consequently, from (1), a : b : c = 6 : 5 : 4.

CMJ473.922.S932.(K.R.S.Sastry)

A triangle ABC is called self-median if it is similar to the triangle thelengths of whose sides are the lengths of the medians in ABC. Let G be thecentroid of a non-isosceles triangle ABC. Prove that the bisectors of ABCand AGC intersect on the side AC if and only if ABC is self - median.

CMJ477.923.S933.(N.Schaumberger)

Let p denote the perimeter of ABC. Prove that

a sinA+ b sinB + c sinCp

≥ 3√

sinA sinB sinC.

CMJ480.923.S933.(I.Sadoveanu)

Let A′, B′, C ′ denote the feet of the altitudes in the triangle ABC, lying onthe lines BC,CA,AB respectively. Show that AC ′ = BA′ = CB′ if andonly if ABC is an equilateral triangle.

CMJ482.924.S934.(I.Sadoveanu)

Let A′, B′, C ′ denote the points of tangency of the sides of triangle ABC toits incircle, opposite A, B and C respectively. Let M be any point on theincircle. Prove that

d(M,AB) · d(M,BC) · d(M,CA) = d(M,A′B′) · d(M,B′C ′) · d(M,C ′A′).

CMJ486.925.S935.(J.Sarkar)

Let X,Y,Z be points on sides BC,CA,AB respectively of ABC such thatAZ : ZB = BX : XC = CY : Y A. Suppose that AX intersects BY at P ,BY intersects CZ at Q and that CZ intersects AX at R. If P , Q, R do notcoincide, prove that

AR : RP : PX = BP : PQ : QY = CQ : QR : RZ.



Let X be a fixed point within a given circle. Let A, B, C and D be variablepoints on the circle such that AC and BD are perpendicular chords throughX. For each X, find the maximum and minimum of (a) the area of thequadrilateral ABCD, and (b) the sum of the lengths of AC and BD.

CMJ493.931.S941.(K.R.Sastry)

Let ABCDE be a convex, affine regular pentagon in which each side isparallel to a diagonal. Let P,Q,R, S, T be points on the sides CD, DE,EA, AB, BC respectively, so that AP , BQ, CR, DS, ET concur at X.Describe the set of points X for which

XP

PA+XQ

QB+XR

RC+XS

SD+XT

TE

is constant.


Let ABC be an integer-sided right triangle, with C = 90. The gcd ofthe lengths of its sides is one. Prove or disprove: its semiperimeter is atriangular if and only if either a and c or b and c are consecutive integers.

See also CMJ535.944.

CMJ503.933.S943.(L.R.Bragg and J.W.Grossman)

How many triangles are there with integral sides of length at most n ?See also Crux 19.


Let a, b, c, d be the lengths of the successive sides of a quadrilateral, and leta be the maximum length.

(a) Prove that if b ≥ d, or if b ≤ d and c ≤ d,

bc

a+cd

b+da

c+ab

d≥ a+ b+ c+ d.

(b) Show that the inequality may fail without the given restrictions onb, c and d.



Find all Pythagorean triangles whose sides are, respectively, n−, (n + 1)−and (n+ 2)−gonal numbers all of the same rank.

CMJ511.935.S945.(Zhang Zaiming)

Let ai, bi, ci, i = 1, 2 be the length of the sides of triangles with area i.Prove that

a21a

22 + b21b

22 + c21c

22 ≥ 1612,

where equality holds if and only if the two triangles are equilateral.

CMJ515.935.S945.(L.Hoehn)

Let ABCD be a quadrilateral with AB = b, BC = c, CD = d, and DA = a.Prove that ABCD is a parallelogram if and only if

ab cosA+ bc cosB + cd cosC + da cosD = 0.

CMJ517.941.S951.(K.S.Sadati)

Let a, b, and c (a < b < c) denote the lengths of the sides of a triangle oppo-site the interior angles A, B, C. Prove that if a2, b2, c2 are in arithmeticalprogression, so are cotA, cotB, cotC.

CMJ529.943.S952. (J.Fukuta)

Let A, B, C, and D be consecutive vertices of a rectangle. Find the locusof points P interior to ABCD such that PA · PC + PB · PD = AB · BC.

CMJ531.944.S954. (Klamkin and A.Liu)

IfA,B,C andD are consecutive vertices of a quadrilateral such that DAC =55 = CAB, ACD = 15, and BCA = 20, determine ADB.

CMJ535.944.S954. (H.Sedinger)

Prove that thee exist an infinite number of right triangles with sides ofinteger lengths a, b, c such that b > a, c > b+ 1, the gcd of a, b, c is 1, andthe semiperimeter is a triangular number.

See also CM501.933.S943.


CMJ537.945.S955. (H.Gulicher)

In P1P2P3 a cevian through P1 cuts side P2P3 in point Q1. Let K ∈P1Q1, (K = P1), S2 ∈ P1P3, and R3 ∈ P1P2 be such that S2K//P1P2, andR3K//P1P3. Prove that P1Q1, S2K, and R3K are concurrent if and only if

|P1S2||P1P3| ·

|P1P2||P1R3| ·

|P3Q1||P2Q1| = 1.

CMJ538.945.S955. (M.S.Klamkin)

Determine the maximum area of the quadrilateral with consecutive verticesA, B, C, and D if A = α, BC = b and CD = c are given.

CMJ539.945.S955. (N.Juric)

In how many ways can three vertices of an n−dimensional cube be chosenso that the chosen vertices form an equilateral triangle ?

CMJ541.951.S961. (V.Oxman)

Let E be an ellipse in the plane with foci F1 and F2; and let L denote a linein the plane that does not intersect E. Let A be any point on L (exceptthe point of intersection of L with the line through F1 and F2). Construct,using only an unmarked straightedge, a point B on L and a point C on Eso that an ellipse with foci A and B is tangent to E at C.

CMJ543.951.S961. (K.R.S.Sastry)

A parallelogram is called self-diagonal if its diagonals are proportional tothe sides. Let ABCD be a parallelogram in which AB > BC, angle A isacute, E is the midpoint ofAB, and F is chosen so that CEDF is also aparallelogram. Prove that ABCD and CEDF are congruent if and only ifboth are self-diagonal.


Let a, b, c, d be positive real numbers.(a) Prove that a + b > |c − d| and c + d > |a − b| are necessary and

sufficient conditions for there to exist a convex quadrilateral that admits acircumcircle and whose side lengths, in cyclic order, are a, b, c, d.

(b) Find the radius of the circumcircle.



In ABC, let the angles at B and C be acute. Suppose the altitudes fromA intersects BC at D, and let E and F denote the points of intersectionof AD with the bisectors of the angles at B and C respectively. Provethat if BE = CF , then triangle ABC is isosceles. Solution. BE = AD

tan B ·1

cos B2

= AD · 1−2s2

2s(1−s2) , where s := sin B2 . Similarly, CF = AD · 1−2t2

2t(1−t2) , with

t := sin C2 . If BE = CF , then

2s(1 − s2)(1 − 2t2) = 2t(1 − t2)(1 − 2s2),s− s3 − 2st2 + 2s3t2 = t− t3 − 2s2t+ 2s2t3

(s − t) − (s3 − t3) + 2st(s− t) + 2s2t2(s− t) = 0(s − t)[1 − (s2 + st+ t2) + 2st+ 2s2t2] = 0(s − t)[1 − s2 − t2 + st+ 2s2t2] = 0(s − t)[(1 − s2)(1 − t2) + st(1 + st)] = 0.

Since B and C are acute, 0 < s, t < 1. The factor (1 − s2)(1 − t2) +st(1 + st) = 0. We must have s = t. This means B

2 = C2 , and B = C. The

triangle is isosceles.

CMJ548.952.S962. (M.Golomb)

Suppose P is a parallelogram with sides a ≥ b and obtuse angle θ.(i) Prove that there exists a square Q in which P is inscribed if and only

if sin θ + cos θ ≥ ba .

(ii) Prove that Q is unique if and only if P is not a square.


In a triangle with integral sides and integral area (a Heronian triangle) provethat a median and a side cannot be of the same length.

CMJ553.953.S963. (M.Golomb)

Given a regular n−gon with center C, vertices A1, A2, . . . , An and a line Lin the plane of the polygon, let pk be the length of the projection of CAk

onto L. Show that for each integer m, 0 < 2m < n, the sum∑n

k=1 p2mk does

not depend on L.



Prove that for any triangle ABC, there exists one and only one set of pointsD, E, F satisfying:

(a) D lies on side BC, E lies on side CA, and F lies on side AB;(b) EA+AF = BC; FB +BD = CA; DC +CE = AB; and(c) AD, BE, and CF are concurrent.Solution. Denote by a, b, c the lengths of the sides of the triangle:

BC = a, CA = b, and AB = c, and by s = 12(a+ b+ c) the semi-perimeter.

First note that if D, E, and F are respectively the points of contact ofthe sides BC, CA, and AB with the excircles of the triangle on the oppositesides of A, B, and C, it is easy to establish

EA = BD = s− c; FB = CE = s− a; DC = AF = s− b.

Condition (b) is clearly satisfied. Also, AD, BE, CF are concurrent byCeva’s theorem.

AF

FB· BDDC

· CEEA

=s− b

s− a· s− c

s− b· s− a

s− c= 1.

The intersection of AD, BE and CF is usually called the Nagel point ofthe triangle.

Now, we show that this is the only set of points satisfying the conditions(a), (b), (c). Let D′, E′, and F ′ be a set of points satisfying the sameconditions. Suppose AF ′ = AF + ε for some ε. Then

E′A = s− c− ε, BD′ = s− c+ ε;F ′B = s− a− ε, CE′ = s− a+ ε;D′C = s− b− ε, AF ′ = s− b+ ε.

If AD′, BE′, and CF ′ are to be concurrent, then Ceva’s theorem requires

AF ′

F ′B· BD

′

D′C· CE

′

E′A= 1.

This means

(s− a+ ε)(s− b+ ε)(s − c+ ε) = (s− a− ε)(s− b− ε)(s − c− ε),ε[ε2 + (s − a)(s − b) + (s − b)(s − c) + (s− c)(s − a)] = 0.

Since (s− a)(s− b) + (s− b)(s− c) + (s− c)(s− a) > 0. This requires ε = 0.This means the points D′, E′, and F ′ coincide respectively with the pointsD, E, and F .


CMJ564.955.S965. (C.A.Minh)

Let ABCD be a convex quadrilateral. Show that if tanA+tanB+tanC+tanD = 0, then ABCD is either a cyclic quadrilateral or a trapezoidal.


Find the dimensions of all rectangular boxes with sides of integral lengthssuch that the volume is numerically equal to the sum of the lengths of theedges plus the surface area.

CMJ570.961.S971. (M.S.Klamkin)

In ABC the angle bisectors of angles B and C meet the altitude AD atpoints E and F , respectively. IF BE = CF , prove that ABC is isosceles.

CMJ574.962. (R.Patenaude)

Describe the locus of the foci of all ellipses inscribed within a given nonsquarerectangle, i.e., tangent to all four sides of the rectangle.


A convex heptagon A1A2A3A4A5A6A7 is such that the angle at A1 is 90

and AiAj//AkAm if i+ j ≡ k +m (mod 7). Prove that

sin 2A2

sin 2A5=A5A

26

A2A23

.

CMJ578.963.S973. (R.Patenaude)

Determine the lengths of the sides of a triangle with the properties that (i)the sides have integral length and (ii) one angle is twice as large as another.

CMJ583.964. (M.S.Klamkin)

A known property of a parabola is that if tangents are drawn at any twopoints P and Q of the curve, then the line from the point of intersection ofthe tangents and parallel to the axis of the parabola bisects the chord PQ.Does this property characterize the parabola? That is, if a curve has theabove property where the line is drawn parallel


CMJ585.965. (K.R.S.Sastry)

The sides of triangle ABC are relatively prime natural numbers. The inter-nal angle bisector of the angle at A meets BC at D. If BD = AC, provethat (i) AB is a square, and (ii) AD is not an integer. Solution. There isonly one such triangle ABC, with BC = 3, AC = 2 and AB = 4. Thebisector AD has length

√6, and BD = 2, CD = 1.

To justify this, we denote the lengths of BC, AC, and AB by relativelyprime natural numbers a, b, c respectively. Since BD = b, we have CD =a− b. Now, the bisector AD divides BC in the ratio BD : CD = AB : AC.From this, b2 = (a− b)c.

Suppose a and b have a common prime divisor p, with ph, pk the highestpowers dividing a and b respectively. If h = k, then pmin(h,k) is the highestpower of p dividing a − b. Since 2k − min(h, k) > 0, p also divides c,contrary to the assumption that a, b, and c are relatively prime. It follows

that a− b = gcd(a, b)2. From this, c =(

bgcd(a,b)

)2.

a− b must be divisible by p2k.We claim that a − b = 1. If not, every prime divisor of a − b must

divide b, and would be a common divisor of a and b. This is contrary to theassumption that a, b (and c) are relatively prime. It follows that c = b2 is asquare.

Indeed, a = b + 1. By the triangle inequality, b2 = c < a + b = 2b + 1.This is possible only for b = 1, 2. If b = 1, the triangle is degenerate. Fromb = 2, we have a = 3 and c = 4, as claimed.

The length of the bisector AD is computed from the formula AD2 =bc(1 − ( a

b+c)2) to be

√6.

CMJ586.965. (K.R.S.Sastry)

The sides BC, CA, AB of triangle ABC are extended to the points R, P , Q,respectively, so that CR = AP = BQ. Prove that if PQR is equilateral,then so is ABC.

CMJ595.971.S981. (J.B.Romero Marquez)

In a right triangle whose sides are a, b, and c (with a ≤ b < c), evaluate

limb→a

mb −ma

θb − θa,


where ma, mb, θa and θb are the lengths of the medians and the anglebisectors meeting sides a and b respectively.

Solution. We solve a slightly more generally problem. Let a and cbe fixed, and r := c

a . The medians ma and mb are given by Apollonius’Theorem:

m2a =

14(2b2 + 2c2 − a2) and m2

b =14(2c2 + 2a2 − b2).

It follows that

limb→a

m2b −m2

a

a− b= lim

b→a

34(a+ b) =

32a.

Since limb→a(mb+ma) = limb→a

√2c2 + 2a2 − b2 =

√2c2 + a2 = a

√2r2 + 1,

we havelimb→a

mb −ma

a− b=

3a2√

2c2 + a2=

32√

2r2 + 1. (5)

The angle bisectors θa and θb are given by

θ2a = bc

(1 −

(a

b+ c

)2)

and θ2b = ca

(1 −

(b

c+ a

)2).

Here,

θ2b − θ2

a

a− b=

c(a+ b+ c)(a2b+ ab2 + 3abc+ ac2 + bc2 + c3)(a+ c)2(b+ c)2

,

limb→a

θ2b − θ2

a

a− b=

c(c+ 2a)(c3 + 2ac2 + 3a2c+ 2a3)(c+ a)4

=c(c+ 2a)(c2 + ac+ 2a2)

(c+ a)3

=ar(r + 2)(r2 + r + 2)

(r + 1)3.

Since

limb→a

(θb+θa) = 2

√√√√ca(

1 −(

a

a+ c

)2)

= 2a

√√√√r(

1 −(

1r + 1

)2)

=2arr + 1

√r + 2,

we have

limb→a

θb − θa

a− b=r2 + r + 22(r + 1)2

√r + 2. (6)

Combining (1) and (2), we have


limb→a

mb −ma

θb − θa=

3(r + 1)2

r2 + r + 21√

(r + 2)(2r2 + 1).

For a right triangle r = ca =

√2, we have

limb→a

mb −ma

a− b=

32√

5≈ 0.67082 . . . ,

limb→a

θb − θa

a− b=

√17 − 23√

2≈ 0.858221 . . .

limb→a

mb −ma

θb − θa=

314

√15(34 + 23

√2) ≈ 0.78164 . . . .

CMJ 599.972.S982. (Juan-Bosco Romero Marquez)

Let ABC be a triangle with acute angles at B and C. Let H be the footof the perpendicular from A to BC; let D be the foot of the perpendiuclarfrom H to AB; let E be the foot of the perpendiuclar from H to AC; letP be the foot of the perpendiuclar from D to BH; and let E be the foot ofthe perpendiuclar from E to HC. Prove that the angle t A is a right angleif and only if AH = DP +EQ.

CMJ 604.973.S983. (J.Fukuta)

The quadrilateral PQRS is inscribed in a convex quadrilateral ABCD suchthat P , Q, R, S are on the sides AB, BC, CD, DA respectively. Let U andV be arbitrary points on diagonals AC, BD respectively, and let A′, B′, C ′,and D′ be the points of intersection of UB and PQ, V C and QR, UD andRS, and V A and SP respectively. Prove that the value of the expression

PA′

A′Q· QB

′

B′R· RC

′

C ′S· SD

′

D′P

where now PA′ denotes the length of the segment PA′ etc), is independentof the location of points U and V .

CMJ 613.975.S985. (M.S.McClendon)

Let T be an isosceles triangle with congruent sides of length k and inradius1. Suppose that these conditions determine T uniquely. Find the length ofthe altitude on the third side of T .


CMJ 624.982.S992. (Harry Sedinger)

LetC be a point on the line segment AB with |AC| = a > 0 and |CB| =b > 0. Let r be a ray beginning at B and making an angle θ with AB,0 < θ < π. Show that there is a point P on r that maximizes angle APC,and show htat hte distance |BP | is independent of the choice of θ.

CMJ 629.983.S993. (D.Beran)

In triangle ABC, the angle bisectors of angles B and C meet the medianAD at points E and F respectively. If BE = CF , prove that triangle ABCis isosceles.

CMJ 635.984.S994. (S. Zimmermann)

Consider a circle of radius r and an interior point P that is p units from thecenter of the circle.

(a) Show that, for any pair of chords of the circle that intersect P atright angles, the sum of the squares of their lengths is always the same.

(b) Find this sum as a function of r and p.

CMJ 650.992. (K.Korbin)

Suppose that ABC is an equilateral triangle with side s and that D is a pointbetween B and C. Let r1 and r2 dentoe the radii of the circles inscribed intriangles ABD and ACD respectively. Express s as an explicit function ofr1 and r2.

CMJ 664.995. (J. Fukuta)

In a triangle ABC with incenter I, let a, b, c be the lengths of the sides BC,CA, AB respectively. Let [O, r] denote the circle with center O and radiusr. Let U be the radical center of [A, a], [B, b] and [C, c], and let V be theradical center of [A, b+ c], [B, c+ a] and [C, a+ b]. Prove that the points I,U , V are collinear and that I is the midpoint of UV .

This is the same as AMM 10734.994 (van Lamoen).


Geometry Problems in

Pi Mu Epsilon Journal

1949 – 1998

PME1.49F.S51S.(Leo Moser)

Prove the following construction for finding the radius of a circumference.With any point O on the circumference as center and any convenient radiusdescribe an arc PQR, cutting the given circumference in P and Q. WithQ as center and the same radius describe an arc OR cutting PQR in R, Rbeing inside the circumference. Join P and R, cutting the given circumfer-ence in L. Then LR is the radius of the circumference. (This is known asSwale’s construction, and is probably the simplest solution of the problemyet discovered.)

PME3.49F.S50S.()

The lengths of the sides of a triangle are the roots of the cubic equationax3 + bx2 + cx + d = 0. Find the area of the triangle. Solution. Thesemiperimeter is s = − b

2a . Putting x = s− y = −(y + b2a), we have

−a(y +b

2a)3 + b(y +

b

2a)2 − c(y +

b

2a) + d = 0;

−(2ay + b)3 + 2b(2ay + b)2 − 4ac(2ay + b) + 8a2d = 0;8a3y3 + 4a2by2 − 2a(b2 − 4ac)y − (b3 − 4abc+ 8a2d) = 0.

If x1, x2, x3 are the sides of the triangle, the roots of this equation are s−x1,s− x2 and s− x3. It follows that

s(s− x1)(s− x2)(s− x3) =b3 − 4abc+ 8a2d

8a3

and

=1

4a2

√−b(b3 − 4abc+ 8ad) =

14a2

√b(4abc− b3 − 8a2d).

Note that if we assume a > 0, then b and d are negative, and c is positive.


PME4.49F.S50S.(Leo Moser)

Towards the bottom of p.268 of Cajori’s A history of Mathematics (1926)we find the following statements: “Nepoleon proposed to the French mathe-maticians the problem, to divide the circumference of a circle into four equalparts by the compasses only. Mascheroni does this by applying the radius 3times to the circumference; he obtains the arcs AB, BC, CD; then AD is adiameter; the rest is obvious. Show how the “obvious” part of the problemmay be accomplished.

Solution. (John A. Dyer, University of Alabama) Napoleon’s problemmay be stated as follows: By use of compass only find the side of a squareinscribed in a given circle O, radius R, given also the consecutive points A,B, C, D on the circumference such that AB = BC = CD = R, and AD isa diameter.

Now, with A as center and radius AC, draw circle A. Similarly, withD as center and radius DB, draw circle D. Circles A and D intersect atpoints, say E and E′. Then OE is the required side of the inscribed square.

Comment. It is not clear how the vertices of the square can be marked on thegiven circle. Here is a slight variation that works. Given O(A). ConstructA(O) to intersect O(A) at B; B(O) to intersect A(O) at C; C(A) to intersectA(O) at D. Here, OD is a diameter of A(O). Let E be an intersection ofthe circles O(C) and D(B). Construct A(E) to intersect the circle O(A) atP , Q. A, P , Q are three vertices of the square. The fourth one is easy todetermine.

PME6.49F.S52S,52F,55F.(Trigg)

Starting with a straight edge, closed compasses, and two straight line seg-ments a and b, construct the harmonic mean of a and b in the least number ofoperations. Changing the opening of the compasses, drawing a circle or thearc of a circle, and drawing a straight line are each considered an operation.

The first solution, by Trigg, has 10 operations. The second solution,by Bankoff, has 9 operations. The third solution, also by Bankoff, has 8operations.

PME8.49F.S55F.(R.T.Hood)

Consider the stereographic projection of a sphere onto a plane tangent to itat its south pole S, the center of projection being the northpole N . Provethat every great circle on the sphere not passing through N is mapping into


a circle whose center is on the line through N which is perpendicular to theplane of the great circle.

PME9.49F.S55F.()

If the bases of a prismatoid are equal in area, then so are the sectionsequidistant from the midsection.

PME14.50S.S53S.(Trigg)

(a) How may a sealed envelope be folded into a rectangular parallelipiped ifoverlapping is permitted ?

(b) What is the maximum volume so obtainable in terms of the edges aand b of the envelope ?

(c) What must be the relative dimensions of the envelope in order toyield a cube ?

(d) What will be the volume of the cube ?

PME16.50S.S51F.(W.J.Jenkins)

Given a circle and two exterior points not in a straight line with the center.Construct a circle passing through these two points and dividing the givencircle into two equal arcs.

PME18.50S.S51F.(L.J.Burton)

Points A1, B1, C1 are chosen on the sides BC, CA, AB of triangle ABCsuch that AC1 = 1

2C1B, BA1 = 12A1C, CB1 = 1

2B1A. The lines AA1, BB1,CC1 determine a triangle A2B2C2. Show that the area of A2B2C2 is oneseventh of the area of ABC.

PME23.50F.S51S.(R.Dubisch)

If in a triangle with sides a, b and c, we have c ≥ b, c ≥ a, find k such thatc2 = ka2 + b2.

PME24.50F.S51S.(P.J.Schillo)

If θn is the angle opposite the side of length 4n2 in the integer right trianglewith sides 4n2, 4n4 − 1 and 4n4 + 1, where n is any positive integer, show


that

limn→∞

∞∑i=1

θi

is a right angle.

PME34.51F.S52S.(J.S.Frame)

For what values of k are the following twelve points (0,±k,±1), (±1, 0,±k),(±k,±1, 0), the vertices of a regular icosahedron ?

PME38.52S.S52F(Trigg)

In the triangle ABC, AD is a median. Prove that if AMMD = p

q , then CM

extended divides AB in the ratio p2q .

PME42.52S.S52F.(M.Stover)

Prove that the volume of a tetrahedron determined by two line segmentslying on two skew lines is unaltered by sliding the segments along their lines(but leaving their lengths unaltered).

PME43.52S.S52F.(P.W.Gilbert)

Four solid spheres lie on top of a table. Each sphere is tangent to the otherthree. If three of the spheres have the same radius R, what is the radius ofthe fourth sphere ?

PME51.52F.S53F.(Trigg)

Suppose D is the foot of the altitude from C, the vertex of the right anglein the triangle ABC. Show that the area of the triangle determined by theincenters of triangles ABC, ADC and BDC is (a+b−c)3

8c .

PME54.S53S.S53F.(F.L.Miksa)

Given a right triangle ABC, with right angle at C, find a point P on ACso that the inscribed circles of the triangles BPC and BAP will be equal.


PME63.54S.S54F.(Bankoff)

State and solve the problem suggest by the following diagram: OA andOB are two perpendicular radii of a quadrant of a circle. Two circles aredrawn inside the incircle of OAB, each touching each other and the incircleinternally. Another circle is drawn tangent to the quadrant and touchingthe incircle of OAB with AB as a common tangent.

PME65.54S.S69F.(M.Schechter)

Prove that every simple polygon which is not a triangle has at least oen ofits diagonals lying entirely inside of it.

See also PME335.74F.

PME66.54S.S54F.(Trigg)

If three circles with radii a, b,c are externally tangent, there are two circleswith radii r, R which touch the three circles. Show that

1r− 1R

= 2(1a

+1b

+1c),

and that1r

+1R

= 4

√a+ b+ c

abc.

PME68.54F.S55F.(Bankoff)

An ellipse of maximum area is inscribed in a given triangle. Show that thearea of the smallest quadrilateral circumscribing this ellipse is less than thegeometric mean and greater than the harmonic mean of the areas of theellipse and the triangle.

PME73.55S.S69F.(V.Thebault)

Construct three circles with given centers such that the sum of the powersof the center of each circle with respect to the other two is the sum.

PME74.55S.S56S.(H.Helfenstein)

Prove that every convex planar region of area π contains two points twounits apart.


PME75.55S.S58S.(Bankoff)

A line parallel to hypotenuse AB of a right triangle ABC passes throughthe incenter I. The segments included between I and the sides AC and BCare designated by m and n. Show that the area of the triangle is given by

mn(m+√m2 + n2)(n+

√m2 + n2)

m2 + n2.


Find the bounding values of the ratio the sides a and c pf a triangle in orderthat the median to one side and the symmedian to the other side may beconcurrent with the internal bisector of the included angle.

PME80.55F.S57S.(H.Helfenstein)

Prove that the circumscribing circles of four triangles determined by fourplanar lines of general position have a common point.

PME87.56S.S57S.(E.P.Starke)

The centroid G of triangle ABC is actually the center of area of ABC.Determine K, the centroid of the triangle considered as being composed ofthree linear segment. Show how to construct K and find some interestinggeometric properties of this point.

PME92.56F.S58S.(Bankoff)

It has been said that algebra is but written geometry and geometry is butdiagramatic algebra. (Sophie Germain, Memoire sur les surfaces elastiques).In the spirit of this quotation, show geometrically that

∞∑n=2

(#1#2

)−1

= 2.


A circle (p) touches the diameter AB of a semicircle (O) in D, and arc ABofthe semicircle in R, (AD < DB). The perpendicular to OR at P cuts thearc RB in S. If RS2 = DB2 −AD2, find the ratio AD

DB .


PME97.58S.S58F.(A.J.Goldman)

Prove that a triangle of area 1π has a perimeter greater than 2.

PME100.58S.S58F. (Bankoff)

A right triangle ABC (AC > CB is inscribed in a semicircle O whosediameter is AB. The radius OS, perpendicular to AB, cuts AC in R, andCD is the altitude upon AB. Find the ratio SO

RO for which triangles ODCand CDB are both Pythagorean.

PME102.58F.S69S.(L.Moser)

Give a complete proof that two equilateral triangles of edge 1 cannot beplaced, without overlap, in the interior of a square of edge 1.

PME106.58F.S61F,62F.(Klamkin)

An equi-angular point of an oval is defined to be a point such that allintersecting chords through the point form equal angleswith the oval atboth points of intersection (on the same side of thechord). It is a knownelementary theorem that if all interior points of an oval are equi-angular,then the oval is a circle.

(a) Show that if one boundary point of an oval is equi-angular, the ovalis a circle.

(b) Determine a class of non-circular ovals containing at least one equi-angular point.

(c) It is conjectured that a non-circular oval can have, at most one equi-angular point.

PME116.59F.S63S.(Klamkin)

Problem 147, due to Auerbach - Mazur, in the Scottisch book of Problems isto show that if a billiard ball is hit from one corner of a billiard table havingcommensurable sides at an angle of 45 with the table, then it will hitanother corner. Consider the more general problem of a table of dimensionration m

n and initial direction of ball of θ = tan−1 1b , (m,n, a, b are integers).

Show that the ball will first strike another corner after an+bm(an,bm) − 2 cushions.

Furthermore, determine which other corner the ball will strike.


PME120.60S.(M.Goldberg)

(a) All the orthogonal projections of a surface of constant width have thesame perimeter. Does any other surface have this property ?

(b) A sphere may be turned through all orientations while remainingtangent to the three lateral faces of a regular triangular prism. Does anyother surface thave this property ?

PME121.60S.S63S.(Klamkin)

Three circular arcs of fixed total length are constructed, each passing throughtwo different vertices of a given triangle, so that they enclose the maximumarea. Show that the three radii are equal.

PME124.60F(H.Kaye),61S(Klamkin,correction),S62F.

Prove the impossibility of constructing the center of a circle with a straight-edge only, given a chord and its midpoint. Construct the center of an ellipsewith a straightedge only, given a chord and its midpoint.

PME129.61S.S61F.(L.Moser)

If R be a regular polyhedron and P a variable point inside or on R, showthat the sum of the perpendicular distances to the faces of R, extended ifnecessary, is a constant.

PME130.61S.S62S.(H.Kaye)

If P is a variable point on the circular arc AB, show that AP + PB is amaximum when P is the mid-point of the arc AB.

PME135.61F.S62F.(T.E.Hull)

Suppose that k points are placed uniformly around the circumference of acircle with unit radius. Show that the product of the distances from anyone point to the others is equal to k, for any k > 1.

PME136.61F.(M.Goldberg)

What is the smallest convex area which can be rotated continuously withina regular pentagon while keeping contact with all the sides of the pentagon?


PME137.61F.S63F.(L.Moser)

Show that the squares of sides 12 , 1

3 , . . . , 1n , . . . can all be placed without

overlap inside a unit square.

PME140.62S.S63F.(M.Goldberg)

What is the smallest area within which an equilateral triangle can be turnedcontinuously through all orientations in the plane.

Editor’s remark: This problem is unsolved, and similar unsolved onesexist for the square and other regular polygons.

PME141.62S.S63F.(D.J.Newman)

Determine conditions on the sides a and b of a rectangle in order that it canbe imbedded in a unit square.

PME148.62F.(Klamkin)

If a convex polygon has three angles of 60, show that it must be an equi-lateral triangle. Solution. If an n − gon, n > 3, has three 60 angles, theremaining n− 3 angles add up to 2n− 4− 2 = 2(n− 3) right angles. Theseangles cannot be all equal, for otherwise, each of them would be a straightangle. Now, it is easy to see that at least one of these angles exceeds 2 rightangles, contradicting the convexity of the polygon. Thus, n = 3, and this isan equilateral triangle.

PME153.63S.S64F.(Klamkin)

Show that the maximum area ellipse which can be inscribed in an equilateraltriangle is the inscribed circle.

PME156.63F.S65S.(K.S.Murray)

If A and B are fixed points on a given circle and XY is a variable diameter,find the locus of point P .


PME161.64S.S67S.(P.Schillo)

It is conjectured that the smallest triangle in area which can cover any givenconvex polygon has an area at most twice the area of the polygon.

PME162.64S.S65F.(Klamkin)

If a surface is one of revolution about two axes, show that it must be spher-ical.

PME165.64F.S66S.(D.J.Newman)

Express cos θ as a rational function of sin3 θ and cos3 θ.

PME166.64F.S70S.(L.Moser)

Show that 5 points in the interior of a 2 × 1 rectangle always determine atleast one distance less than sec 15.

PME167.64F.S66S.(Klamkin)

Given a centrosymmetric strictly convex figure and an intersecting transla-tion of it, show that there is only one common chord and that this chord ismutually bisected by the segment joining the centers.

PME169.65S.S66S.(J.Konhauser)

From an arbitrary point P (not a vertex) of an ellipse lines are drawn throughthe foci intersecting the ellipse in points Q and R. Prove that the line joiningP to the point of intersection of thetangents to the ellipse at Q and R is thenomral to the ellipse at P .

PME170.65S.S66S.(C.S.Venkataraman)

Prove that a triangle ABC is isosceles or right-angled if

a3 cosA+ b3 cosB = abc.


PME172.65F.S66F.(J.Baudhuin)

Given: semicircle O with diameter AB and equilateral triangle PAB; C andD are trisection points of the chord semicircle AB.

Prove: E and F are trisection points of the chord AB.Note: A synthetic proof is desired.

PME174.65F.S66F.(C.S.Venkataraman)

Find the locus of a point which moves such that the squares of the lengths ofthe tangents from it to three coplanar circles are in arithmetic progression.


Rr2 ≥ 2s2

r1r2r3.

PME178.66S.S67S.(K.S.Murray)

Show that the centroid of ABC coincides with the centroid of A′B′C ′,where A′, B′, C ′ are the midpoints of BC, CA and AB respectively.

Generalize to higher dimensions.

PME180.66S.S67S.(R.C.Gebhart)

In the figure, AB = AC and ABC = 90.The arcs are both circular withthe inner one being tangent to AB at A and BC at C. Determine the areaof the cresent.

PME181.66S.S67F.(D.W.Crowe and Klamkin)

Determine a convex curve circumscribing a given triangle such that(1) the area of the four regions (3 segments and a triangle) formed are

equal, and(2) the curve has a minimum perimeter.

PME187.67S.S68S.(R.C.Gebhart)

A semicircle ACB is constructed on a chord AB of a unit circle. Determinethe chord AB such that the distance from O to C is a maximum.



If A,B,C,D,E, F and G denote the consecutive vertices of a regular hep-tagon, show that CD is equal to the harmonic means of AC and AD.

PME191.67S.(Rabinowitz)

Let P and P ′ denote points inside rectangles ABCD and A′B′C ′D′ respec-tively. If PA = a + b, PB = a + c, PC = c + d, PD = b + d, P ′A′ = ab,P ′B′ = ac, P ′C ′ = cd, prove that P ′D′ = bd.


Mathematics Magazine (January 1963), p.60, contains a short paper byDov Avishdom, who asserts without proof that in the adjoining diagram,AN = NC + CB. Give a proof.

PME198.67S.S68F.(Rabinowitz)

A semi-regular solid is obtained by slicingoff sections from the corners of acube. It is a solid with 36 congruent edges, 24 vertices and 14 faces, 6 ofwhich are regular octagons and 8 are equilateral triangles. If the length ofan edge of this polytope is e, what is its volume ?



In a right triangle, find angle HIO given that HIO is isosceles.

PME203.68S.(Rabinowitz)

Let P denote any point on the median AD of ABC. If BP meets AC atE and CP meets AB at F , prove that AB = AC if and only if BE = CF .

PME205.68F.S69F.(C.S.Venkataraman)

ABC and PQR are two equilateral triangles with a common circumcenterbut different circumcircles. PQR and ABC are in opposite senses. Provethat AP , BQ and CR are concurrent.



Three equal circles are inscribed in a semicircle as shown in the adjoiningdiagram. How is this figure related to one of the better known properties ofthe sequence of Fibonacci numbers ?

PME211.68F.S69F.(L.Barr)

It is known that the sum of the distances from I to the vertices cannotexceed the combined distances from the orthocenter to the vertices. (Amer.Math. Monthly, E1397(1960)). Show that the reverse inequality holds fortheir products, namely, AH ·BH · CH ≤ AI · BI · CI.

PME213.69S.S70S,84S.(G.Wulczyn)

Prove that a triangle is isosceles if and only if it has a pair of equal ex-symmedian.

This is false. See also Crux Math. 9 (1983) p.181.See akso MG637 for the corresponding involving symmedians.


In an acute triangle ABC whose circumcenter is O, let D,E,F denote themidpoints of sides BC, CA, AB and let P , Q, R denote the midpoints ofthe minor arcs BC, CA, AB of the circumcircle. Show that

DP + EQ+ FR

OB +OD +OC +OE +OA+OF=

sin2 A2 + sin2 B

2 + sin2 C2

cos2 A2 + cos2 B

2 + cos2 C2

.


A transverse common tangent of two circles meets the two direct commontangents in B and C. Prove that the feet of the perpendiculars from Band C on the line of centers ar a pair of common inverse points of both thecircles.

PME220.69S.S70F,71S.(Pedoe)

(a) Show that there is no solution of the Apollonius problem of drawingcircle to touch three given circles which has only seven solutions.

(b) What specializations of the three circles will produce 0,1,2,3,4,5,6distinct solutions ?


PME222.69F.S70F.(Garfunkel)

In an acute triangle ABC, angle bisector BT1 intersects altitude AH1 in D.Angle bisector CT2 intersects altitude BH2 in E, and angle bisector AT3

intersects altitude CH3 in F . Prove that

DH1

AH1+EH2

BH2+FH3

CH3≤ 1.

PME229.69F.S70F.(C.L.Main)

Let A1 and A2 be tangent unit circles with a common external tangent T .Define a sequence of circles recursively as follows:

(1) C1 is tangent to T ,A1 and A2;(2) Ci is tangent to Ci−1, A1 and A2, for i = 2, 3, . . .Find the area of the region ∪iCi.

PME231.69F.S70F.(D.L.Silverman)

(a) What is the smallest circular ring through which a regular tetrahedronof unit edge can be made to pass ?

(b) What is the radius of the smallest right circular cylinder throughwhich the unit edge tetrahedron can pass ?

PME237.70S.S71S.(L.Barr)

The diameter of a semicircle is divided into two segments, a and b, by itspoints of contact with an inscribed circle. Show that the diameter of theinscribed circle is equal to the harmonic mean of a and b.

PME238.70S.(D.L.Silverman)

A necessary and sufficient condition that a triangle exist is that its sidessatisfy the inequalities

a < b+ c, b < c+ a, c < a+ b.

Express these in a single inequality.


PME239.70S.S84F.(D.L.Silverman)

A pair of tori having hole-radius = tube radius = 1 are linked.(a) What is the smallest cube into which the tori can be packed ?(b) What convex surface enclosing the linked tori has the smallest volume

?(c) What convex surface enclosing the linked tori has the smallest area

?(d) What is the locus of points in space equidistant from the two links ?

PME239’.70F.S71F.(Anonymous)

A circle (O) inscribed in a square ABCD, (AB = 2a), touches AD at G,DC at F , and BC at E. If Q is a point on DC and P a point on BC suchthat GQ is parallel to AP , show that PQ is tangent to the circle (O).


m2am

2b +m2

bm2c +m2

cm2a = 9

16(a2b2 + b2c2 + c2a2).

PME243.70F.S71F.(A.E.Neuman)

Provide a geometrical proof for the well known relation

π

4= arctan

12

+ arctan15

+ arctan18.

PME247.70F.S71F.(A.E.Neuman)

Construct diagrams illustrating four (or more) different theorems charac-terised by the relation

AZ ·BX · CY = |AY · BZ · CX|.The diagrams given by the proposers invoke the following theorems:

Menelaus, Ceva, Desargues, and Morley.

PME254.71S.S72S.(A.E.Neuman)

In the adjoining diagram, CD is a half-chord perpendicular to the diameterAB of a circle (O). The circles on diameters AC and CB are centered on O1

and O2 respectively. The rest of the figure consists of consecutively tangentcircles inscribed in the horn - angle and in the segment as shown. If the twoshaded circles are equal, what is the ratio of AC to AB ?


PME256.71S.S72S.(R.S.Luthar)

ABCDE is a pentagon inscribed in a circle (O) with sides AB, CD and EAequal to the radius of (O). The midpoints of BC and DE are denoted by Land M respectively. Prove that ALM us ab equilateral triangle.

PME257.71S.S72S.(M.Louder and R.Field)

If x, y, z are the sides of a primitive Pythagorean triangle with z > x > y,can x and x− y be the legs of another Pythagorean triangle ?

PME259.71F.S72F.(J.Bender)

Prove that the product of the eccentricities of two conjugate hyperbolas isequal to or greater than 2.

PME260.71F.S72F.(Erros)

Given n points in the plane, what is the maximum number of triangles youcan form so that no two triangles have an overlap in area ?


If A+B + C = 180, prove that

cosA

2+ cos

B

2+ cos

C

2≥ sinA+ sinB + sinC.

PME270.72S.S73S.(Carlitz)

cotA

2+cot

B

2+cot

C

2≥ 3(tan

A

2+tan

B

2+tan

C

2) ≥ 2(sinA+sinB+sinC).


Twelve toothpicks can be arranged to form four congruent equilateral tri-angles. Rearrange the toothpicks to form ten triangles of the same size.

PME277.72S.S73S.(A.E.Neuman)

According to Morley’s theorem, the intersections of the adjancent internalangle trisectors of a triangle are the vertices of an equilateral triangle. Ifthe configuration is modified so that the trisectors of one of the angles are


omitted, as shown in the diagram, show that the connector DE of the twointersections bisects the angle BFC.

PME288.72F.S73F.(Bankoff and A.E.Neuman)

If A+B + C = π, show that(1) sin 2A+ sin 2B + sin 2C ≤ sinA+ sinB + sinC;(2) sin 2A+sin 2B+sin 2C ≤ sinA+sinB+sinC+sin 3A+sin 3B+sin 3C;equality holding if and only if A = B = C.


How may a square card be folded into a tetrahedron ? What is the volumeof the tetrahedron in terms of the side of the square ?

PME292.73S.S74S,75S.(Garfunkel)

If perpendiculars are constructed at the points of tangency of the incircleof a triangle and extended outward to equal lengths, then the join of theirendpoints form a triangle perspective with the given triangle.


Show that ABCD is a square.


Determine the equation of a regular dodecagon (the extended sides are notto be included).


It can be shown without difficulty that if the opposite anlges of a skewquadrilateral are equal in pairs, the opposite sides are also equal in pairs.If two opposite sides of a skew quadrilateral are equal and the other twounequal, is it possible to have one pair of opposite angles equal ?

PME302.73S.S74S.(D.L.Silverman and A.E.Neuman)

A tapestry is hung on a wall so that its upper edge is a units and its loweredge b units above the observer’s eye level. Show that in order to obtain the


most favorable view the observer should stand at the distance√ab from the

wall.


In an acute triangle ABC, AF is an altitude and P is a point on AF suchthat AP = 2r, where r is the inradius of the triangle. If D and E are theprojections of P upon AB and AC respectively, show that the perimeter oftriangle ADE is equal to that of the triangle of least perimeter that can beinscribed in triangle ABC.


On opposite sides of a diameter of a circle with radius a+ b two semicircleswith radii a and b form a continuous curve that divides the circle into twotadpole-shaped parts.

(a) Find the angle that the join of the centroids of the two componentparts makes with the given diameter of the circle.

(b) For what ratios a : b does the continuous curve pass through one ofthe centroids ?

(c) When a = b, find the moment of inertia of one of the componentareas about an axis through its centroid and perpendicular to its plane.

PME313.73F,74F(corrected),S75S.(Klamkin)

Give an elementary proof that

(1 + 8 cos2A)(1 + 8 cos2B)(1 + 8 cos2 C) ≥ 64 sin2A sin2B sin2 C,

where A,B,C are the angles of an acute triangle ABC.

PME314.74S.S75S.(J.A.H.Hunter)

Show thatsin2 45 − sin2 15

sin2 30 − sin2 10=

sin 80

sin 30.

PME316.74S.S75S.(Z.Katz)

Which is greater:

2 arctan(√

2 − 1) or 3 arctan14

+ arctan599

?



A rectangle ADEB is constructed externally on the hypotenuse AB of aright triangle ABC. The line CD and CE intersect the line AB in thepoints F and G respectively.

(a) If DE = AD√

2, show that AG2 + FB2 = AB2.(b) If AD = DE, show that FG2 = AF ·GB.

PME319.74S.S75S.(M.S.Longuet-Higgins)

Let A′, B′, C ′ be the images of an arbitrary point in the sides BC. CA andAB of triangle ABC. Prove that the 4 circles AB′C ′, BC ′A′, CA′B′ andABC are all concurrent.

PME320.74S.S75S.(Coxeter)

Prove that the projectivity ABC∧BCD for 4 collinear points is of the period4 if and only if H(AC,BD).

PME322.74S.S75S.(Garfunkel)

It is known that the ratio of the perimeter of a triangle to the sum of itsaltitudes is greater than or equal to 2√

3. (See E1427, (1961) pp.296–297).

Prove the stronger inequality for the internal angle bisectors ta, tb, tc:

2(ta + tb + tc) ≤√

3(a+ b+ c),

equality holding if and only if the triangle is equilateral.

PME323.74S.S75S.(D.L.Silverman)

Call plane curves such as the circle of radius 2, the square of side 4, orthe 6 × 3 rectangle isometric if theri perimeter is numerically equal to thearea they enclose. What is the maximum area that can be enclosed by anisometric curve ?


In a right triangle ABC, A = 60, B = 30, with D,E,F the points oftrisection nearest A,B,C on the sides AB,BC,CA respectively. ExtendCD,AE and BF to intersect the circumcircle (O) at points P,Q,R. Showthat triangle PQR is equilateral.


PME335.74F.S75F.(V.G.Feser)

(a) Show that every simple polygon of n sides, n ≥ 3, has at least n − 3interior diagonals.

(b) SHow that for every n ≥ 3, there existsa simple polygon havingexactly n− 3 interior diagonals.

See also PME65.54S.S69F.

PME336.74F.S75F.(Z.Katz)

On the diameter AB of a semicircle (O) perpendiculars are erected at ar-bitrary points C and D cutting the semi-circumference at points E and Frespectively. A circle (P ) touches the arc of the semicricle and each of thetwo half-chords. Show that PQ, the distance from P to the diameter AB,is equal to the geometric mean of AC and DB.


If R, r and ρ denote the circumradius, the inradius and the orthic triangleradius respectively of an acute triangle, show that r2 ≥ ρR.

PME338.75S.S76S.(H.C.Li)

Let (O)a be a circle centered at O with radius a. Let P , any point on thecircumference of (O), be the center of circle (P ). What is the radius of (P )such that it divides the area of (O) into two regions whose areas are in theratio s : t ?

PME341.75S.S76S,76F.(Garkfunkel)

Prove that the following construction triects an angle of 60. Triangle ABCis a 30−60−90 right triangle inscribed in a circle. Median CM is drawnto side AB and extended to M ′ on the circle. Using a marked straightedge,point N on AB is located such that CN extended to N ′ on the ecircle makesNN ′ = MM ′. Then CN trisects the 60 angle ACM .

PME344.75S.S76S.(J.A.H.hunter)

Three circles whose radii are a, b and c are tangent externally in pairs andare enclosed by a triangle each side of which is an extended tangent of twoof the excircles. Find the sides of the triangles.



The internal angle bisectors of a convex quadrilateral ABCD enclose anotherquadrilateral EFGH. Let FE and GH meet in M and let GF and HE meetin N . If the internal bisectors of angles EMH and ENF meet in L, showthat angle NLM is a right angle.


Angles A and B are acute angles in ABC. If A = 30 and ha, the altitudeissuing from A, is equal to mb, the median issuing from B, find angles Band C.


The edges of a semi-regular polyhedron are equal. The faces consist of eightequilateral triangles and six regular octagons. In terms of the edge e, findthe diameters of the following spheres: (i) the sphere touching the octagonalfaces, (b) the circumsphere, and (c) the sphere touching the triangular faces.

See also PME198.67S.S68F.

PME354.75F.S76F.(A.Bernhart and D.C.Kay)

In ABC with angles less than 60, the Fermat point, defined as that pointwhich minimizes the function f(X) = AX +BX+CX, may be determinedas the point P of concurrence of lines AD,BE,CF , where BCD, ACEand ABF are equilateral triangles constructed externally on the sides ofthe triangle ABC. If R, S and T are the points where PD, PE, and PFmeet the sides of triangle ABC, prove that PD, PE and PF are twice thearithmetic means, and that PR, PS and PT are half the harmonic meansof the pairs of distances (PB,PC), (PC,PA), and (PA,PB) respectively.

PME361.75F.S76F.(C.A.Argila)

Consider any triangle ABC such that the midpoint P of side BC is joinedto the midpoint Q of side AC by the line segment PQ. Suppose R and Sare the projections of P and Q respectively on AB, extended if necessary.what relationship must hold between the sides of the triangle if the figurePQRS is a square.



A diameter AB of a circle (O) passes through C, the midpoint of a chordDE. M is the midpoint of arc AB and the chord MP passes through C.The radius OP cuts the chord DE at Q. The tangent circle (O)1), (O2),(W1), (W2) are drawn as shown. Show that DQ = W1W2.

PME367.76S.S77S.(R.R.Rowe)

A box of unit volume consists of a square prism topped by a pyramid. Findthe side of the square base and heights of prism and pyramid to minimizethe surface area.


Given a triangle ABC with its inscribed circle (I), lines AI, BI, CI cutthe circle in points D,E,F respectively. Prove that AD +BE + CF ≥ 1√

3perimeter of DEF .


In a triangle ABC inscribed in a circle (O), angle bisectors AT1, BT2, CT3

are drawn and extended to the circle. Perpendicualrs T1H1, T2H2, T3H3 aredrawn to sides AC,BA,CB respectively. Prove that T1H1 + T2H2 + T3H3

does not exveed 3R, where R is the circumradius.

PME380.76F.S77F.(V.F.Ivanoff)

Form a square from a quadrangle by bisecting segments and the angles.

PME383.76F.S77F.(N.Schaumberger)

Find a pentagon such that the sum of the squares of its sides is equal tofour times its area.

PME385.76F.S77F.(J.T.Hurt)

Solve sinα = tan(α − β) + cosα tan β.



Show that the volume of Kepler’s Stella Octagula ( a compound of twointerpenetrating tetrahedra) is three times that of the octahedron that wasstellated.


On the sides AB and AC of an equilateral triangle ABC mark the points Dand E respectively such that AD = AE. Erect equilateral triangles on CD,AE and AB, as in the figure, with P,Q,R as the respective third vertices.Show that triangle PQR is equilateral. Also show that the midpoints ofPE,AQ and RD are vertices of an equilateral triangle.

PME390.77S.S78S.(R.Koether and D.C.Kay)

Let the diagonals of a regular n−gon of unit side be drawn. Prove that then − 2 consecutive triangles thus formed which have their bases along onediagonal, their legs along two others or a side, and one vertex in commonwith a vertex of the polygon each have the property that the product of twosides equals the third.

PME394.77S.S78S.(E.Just and B.Kabak)

3(sin2A+ sin2B + sin2 C) − 2(cos3A+ cos3B + cos3C) ≤ 6.

PME398.77S.S78S.(R.S.Field)

Find solutions in integers A = B = C = R and A = B = C = R for thequadrilateral inscribed in a semicircle of radius R, as shown in the diagrambelow. Find also solutons in integer A = B = C = R or prove that noneexist.


Show that

arcsinx− 3

3+ 2arccos

√x

6=π

2, 3 ≤ x ≤ 6.


PME405.77F,78F(corrected).S79F.(N.Schaumberger)

Locate a point P in the interior of a triangle such that the product of thethree distances from P to the sides of the triangle is a maximum.

PME406.77F.S78F.(Erdos)

Let there be given 5 distinct points in the plane. Suppose they determineonly two distances. Is it true that they are the vertices of a regular pentagon?

PME408.77F.S78F.(C.W.Dodge)

Squares are erected on the sides of a triangle, either all externally or allinternally. A circle is centered at the center of each square wotj each radiusa fixed multiple k > 0 of the side of that square. Find k so that the radicalcenter of the three circles falls on the Euler line of the triangle and findwhere on the Euler line it falls.


A point E is chosen on side CD of a trapezoid ABCD, AD//BC), and isjoined to A and B. A line through D parallel to BE intersects AB in F .Show that FC is parallel to AE.

PME410.77F.S78F.(Klamkin)

If x, y, z are the distances of an interior point of a triangle ABC to the sidesBC, CA, AB, show that

1x

+1y

+1z≥ 2r

where r is the inradius of the triangle.

PME412.78S.S79S.(S.W.Golomb)

Are there examples of angles which are trisectible but not constructible ?That is, can you find an angle α which is not constructible with straightedge and compass, but such that when α is given, α

3 can be constructedfrom it with straight edge and compass ?


PME416.78S.S79S.(S.Kim)

Each of the three figures shown above is composed of two isosceles righttriangles, ABC and DBE, where ABC and DBE are right angles,and B is between points A and D.

In (a), points C and E coincide, so that CBEB = 1.

In (b), CBEB = 2.

In (c), CBEB = 3.

Consider each pair of triangles asa single shape and suppose that theareas of the three shapes are equal.

Problem: for each pair of figures, find the minimum number of piecesinto which the first figure must be cut so that the pieces may be reassembledto form the second figure. Pieces may not overlap, and all pieces mut beused in each assembly.

PME417.78S.S79S.(C.W.Dodge)

(a) Prove that the line joining the midpoints of the diagonals of a quadri-lateral circumscribed about a circle passes through the center of the circle.

(b) Let the incircle of triangle touch side BC at X. Prove that the linejoining the midpoints of AX and BC passes through the incenter I of thetriangle.

PME418.78S.S79S.(R.C.Gebhardt)

Find all angles θ such that tan 11θ = tan 111θ = tan 1111θ = tan 11111θ =· · ·.

PME420.78S.S79S.(H.Taylor)

Given four lines through a point in 3-space, no three of the lines in a plane,find four points, one on each line, forming the vertices of a parallelogram.

See also Putnam Competition, 1977, B2.


If F (x, y, z) is a symmetric increasing function of x, y, z, prove that for anytriangle, in which wa, wb, wc are the internal angle bisectors and ma,mb,mc

the medians, we have

F (wa, wb, wc) ≤ F (ma,mb,mc)



PME422.78S.(Garfunkel)

If perpendiculars are erected outwardly at A, B of a right triangle ABC,C = 90), and at M , the midpoint of AB, and extended to points P , Q,and R such that AP = BQ = MR = 1

2AB, show that triangle PQR isperspective with triangle ABC.


Without using its altitude, compute the volume of a regular tetrahedron bythe prismoidal formula.

PME427.78F.S79F.(J.E.Fritts)

If a, b, c, d are integers and u =√a2 + b2, v =

√(a− c)2 + (b− d)2, and

w =√c2 + d2, then√

(u+ v + w)(u + v − w)(u− v + w)(−u+ v + w)

is an even integer. Solution. This is four times the area of the triangle withvertices (a, b), (c, d), and the origin, and is therefore |ad− bc|.

PME428.78F.S79F.(S.W.Golomb)

One circle of radius a may be “exactly surrounded” by 6 circles of radiusa. It may also be exactly surrounded by n circle of radius t, for any n ≥ 3,where

t = a(cscπ

n− 1)−1.

Suppose instead we surround it with n+ 1 circles, one of radius a and n ofradius b, (again n ≥ 3). Find an expression for b

a as a fucntion of n.

PME430.78F.S79F.(J.M.Howell)

Given any rectangle, form a new rectangle by adding a square to the longside. What is the limit of the long side to the short side ?


In a right triangle ABC, with sides a, b and hypotenuse c, show that 4(ac+b2) ≤ 5c2.


PME435.78F.S79F.(D.R.Simonds)

Two non-congruent triangles are “almost congruent” if two sides and threeangles of one triangle are congruent to two sides and three angles of theother triangle. Clearly two such triangles are similar. Show that the ratioof similarity k is such that 1

φ < k < φ, where φ := 12(√

5 + 1) is the goldenratio.Editor’s Note: This old problem is being reopened with the hope of elicitingfresh insights.

PME436.78F.S79F.(C.Spangler and R.A.Gibbs)

P1 and P2 are distinct points on lines L1 and L2 respectively. Let L1 and L2

rotate about P1 and P2 respectively with equal angular velocities. Describethe locus of their intersection.


In times gone by, it was fairly well known that N , the Nagel point of atriangle, is the intersection of the lines from the vertices to the points ofcontact of the opposite escribed circles. In the triangle whose sides areAB = 5, BC = 3 and CA = 4, show that the areas of triangles ABN , CANand BCN are 1,2,3 respectively.

PME438.79S.S80S,80F,81S.(E.Straus)

Prove that the sum of the lengths of alternate sides of a hexagon withconcurrent major diagonals inscribed in the unit circle is less than 4.


Show that the sum of the perpendiculars from the circumcenter of a triangleto its sides is not less than the sum of the perpendiculars drawn from theincenter to the sides to the triangle.


A variable circle touches the circumferences of two internally tangent circles,as shown in the figure.

(a) Show that the center of the variable circle lies on an ellipse whosefoci are the centers of the fixed circles.


(b) Show that the radius of the variable circle bears a constant ratio tothe distance from its center to the common tangent of the fixed circles.

(c) Show that this constant ratio is equal to the eccentricity of the ellipse.

PME448.79S.S80S.(R.R.Rowe)

Analogous to the median, call a line from a vertex of a triangle to a thirdpoint of the opposite side a “tredian”. Then if both tredians are drawn fromeach vertex, the 6 lines will intersect at 12 interior points and divide the areainto 19 subareas, each a rational part of the area of the triangle. Find twotriangles for which each subarea is an integer, one being a Pythagorean righttriangle and the other with consecutive integers for its three sides.

PME450.79F.S80F.(C.W.Dodge)

In triangle ABC, A ≤ B ≤ C. Prove that

s >, = or <√

3(R + r) if and only if B >, = or < 60.


Given two intersecting lines and a circle tangent to each of them, constructa square having two of its vertices on the circumference of the circle and theother two on the intersecting lines.

PME454.79F.S80F.(M.Haste)

The point within a triangle whose combined distances to the vertices is aminimum is known as the Feremat - Toricelli point, designated by T . In atriangle ABC, if AT , BT , CT form a geometric progression with a commonratio 2, find the angles of the triangle.

PME459.79F.S80F.(B.Prielipp)

If (x, y, z) is a Pythagorean triple in which x and z are prime numbers andx ≥ 11, show that 60 divides y.

PME460.79F.S80F.(B.Seville)

The dihedral angle of a cube is 90. The other four Platonic solids have dihe-dral angles which are approximately 7031′43.60′′, 10928′16.3956′′ , 11633′54.18′′,and 13811′22.866′′. How closely can these angles be constructed with


straightedge and compass ? Can good approximations be accomplished bypaper folding ? If so, how ?

PME461.79F.S80F.(D.C.Kay)

(a) A right triangle with unit hypotenuse and legs r and s is used to form asequence of similar right triangles T1, T2, T3, . . . where the sides of T1 are rtimes those of the given triangle, and for n ≥ 1 the sides of Tn+1 are s timesthose of Tn. Prove that the sequence Tn will tile the given triangle.

(b) What happens if the multipliers r and s are reversed ?(c) The art of the Hopi American Indians is known for its zigzag patterns.

The blanket illustrated below is made from a rectangle of (inside) dimensionsa × b, and the zigzag is formed by dropping perpendiculars to alternatingsides of the triangles in the design. Show that the area of the design (shadedportion) is given by the formula a3b+ab3

2a2+4b2.


What is the shortest strip of equilateral triangles of side k that, while remain-ing intact, can be folded along the sides of the triangles so as to completelycover the surface of an octahedron with edge k ?

PME469.80S.S81S.(R.I.Hess)

Start with a unit circle and circumscribe an equilateral triangle about it.Then circumscribe a circle about the triangle and a square abotu the circle.Continue indefinitely circumscribing circle, regular pentagon, circle, regularhexagon, etc.

(a) Prove that there is a circle which contains the entire structure.(b) Find the radius of the smallest such circle.

PME471.80S.S81S.(C.W.Dodge)

Let two circles meet at O and P ,and let the diameters OS and OT of thetwo circles cut the other circle at A and B. Prove that chord OP passesthrough the cneter of circle OAB.


In an acute triangle with angle A = 60, P is a point within the triangle. Dand E are the feet of the cevians through P , from C and B respectively.


(a) If BD = DE = EC, prove that AP = BP = CP .(b) Conversely, if AP = BP = CP , prove that BD = DE = EC.(c) If PBC = PCB = 30, show that BD = DE = EC.


In the accompanying diagram, DC is the radius perpendicular to the diam-eter AB of the semicircle ADB; FG is a half-chord parallel to DC; AF cutsDC in E. Show that the sides of triangle FCG are integers if and only ifDEEC or its reciprocal is an integer.

PME476.80F,(correction 81S).S82S.(Garfunkel)

If A,B,C,D are the internal angles of a convex quadrilateral, then√

3(cosA

2+ cos

B

2+ cos

C

2) ≤ cot

A

2+ cot

B

2+ cot

C

2,

with equality when A = B = C = D = 90.

PME484.80F.S81F.(R.R.Robinson)

In a triangle with base AB and vertex C, secants from A and B to pointsD and E on BC and CA divide the area into four subareas S, T , U andV . In some order of S, T , U , V , the points D and E can be located sothat the subareas are in increasing arithmetical progression, or so that theyare in decreasing arithmetical progression. Find that order and evaluate thesubareas.

PME485.80F.S81F.(R.S.Luthar)

A line l cuts two parallel rays emanating from L andM in A and B re-spectively. A point C is taken anywhere on l. Lines through A and Brespectivley parallel to MC and LC intersect in P . Find the locus of P .


Given an acute triangle ABC with altitudes ha, hb, hc, and medians ma,mb, mc. The points P , Q, R are the intersections of ma and bb; mb and hc;and mc and ha respectively. Show that

AP

PD+BQ

QE+CR

RF≥ 6.

Here, D, E, F are the midpoints of the sides.



In the diagram CD is a half-chord perpendicular to the diameter AB of thesemicircle (O), and the inscribed circle (P ) touches AB in J and the arcDB in K.Show by elementary plane geometry, without using inversion, that

PME495.81S.S82S.(R.Hess)

A regular pentagon is drawn on an ordinary graph paper. Prove that nomore than two of its vertices lie on grid points.

PME496.81S.S82S.(D.Conrad)

P is any point within ABC. If x is the distance from P to BC, show that

PA2 = PH2 + b2 + c2 − 4R2 − ax

2s(b2 + c2 − a2).

PME501.81F.S82F.(R.C.Gebhardt)

A rectangle is inscribed inside a circle. The area of the circle is twice the areaof the rectangle. What are the proportions of the the rectangle ? Answer.4 −√

16 − π2 : π ≈ 0.485.

PME507.81F.S82F.(H.R.Bailey)

A unit square is to be covered by three circles of equal radius. Find theminimum necessary radius.



Given a triangle ABC whose incircle touches the sides BC, CA, AB atL, M , N . Let P , Q, R be the midpoints of the arcs NL, LM and MNrespectively. Form triangle DEF by drawing tangents to the circle at P , Q,R. Prove that the perimeter of DEF ≤ perimeter of ABC.

PME514.82S.S83S.(R.E.Spaulding)

Let A1A2 · · ·An be a regular n−gon each side of length 1. Let Bi be a pointon AiAi+1 such that AiBi = x. Let Ci be the point where AiBi+1 intersectsAi+1Bi+2. Find the area of the regular polygon C1C2 · · ·Cn in terms of nand x.



Given a sequence of concentric circles with a triangle ABC circumscribingthe outermost circle. Tangent lines are drawn from each vertex of ABC tothe next inner circle, forming the sides of triangle A′B′C ′. Tangents arenow drawn from vertices A′, B′, C ′ to the next inner circle and they arethe sides of triangle A′′B′′C ′′ and so on. Prove that the angles of triangleA(n)B(n)C(n) approach π

3 .

PME523.82F.S83F.(Rabinowitz)

Let ABCD be a parallelogram. Eerect directly similar right triangles ADEand FBA outwardly on sides AB and DA (so that ADE and FBA are rightangles. Prove that CE and CE are perpendicular.

PME528.82F.S83F.(A.Wayne)

Call a trio like (19, 24, 35), (15, 29, 34) and (14, 31, 33) a size triplet becausethe three triangles have the same perimeter and the same area. Since thecommon area is least, this is the smallest size triplet. What is the nextlarger size triplet ?

Remark: See also AMME2872.S827. Example of 10 triangles with equalarea and perimeter:

(124700, 830280, 579020), (1246032, 752250, 653718),(1245675, 765765, 640560), (1182675, 1101360, 367965),(1186770, 1093950, 371280), (1206660, 1047540, 397800),(1219920, 1001130, 430950), (1233180, 928200, 490620),(1236495, 901680, 513825), (1246440, 729300, 676260).


In the accompanying diagram, AB = 2r is the diameter of circle (O) andAC = 2r1 the diameter of circle (O1), D is a point on diameter AC, and thehalf - chord DQ perpendicular to AC cuts the circle (O1) at P . The circle(W1) of radius ρ1 and (W2) of radius ρ2 are tangent to circle (O) and (OP1)and touch PQ on opposite sides. Show that ρ1

ρ2= r1

r .


If the exradii satisfy (r1−r2)(r1−r3) = 2r2r3, determine which of the anglesA, B, C is the largest.


PME541.83S.S84S.(Rabinowitz)

A line meets the boundary of an annulus A1 in four points P , Q, R, S withR and S between P and Q. A second annulus A2 is constructed by drawingcircles on PQ and RS as diameters. Find the relationship between the areasof A1 andA2.

PME542.83S.S84S.(H.R.Bailey)

A circle of unit radius is to be covered by three circles of equal radii. Findthe minimum radius required.


Show that a quadrilateral ABCD with sides AD = BC = s and A+ B =120 has maximum area if it is an isosceles trapezoid.

PME550.83F.S84F.(I.R.Hess)

How many different Pythagorean triples have a side or hypotenuse equal to1040 ?


Given a triangle ABC rect equilateral triangles BAP , ACQ outwardly onsides AB and CA. Let R be the midpoint of side BC and let G be thecentroid of triangle ACQ. Prove that triangle PRG is a 30 − 60 − 90

triangle.

PME555.83F.S84F.(R.D.Stratton)

18 toothpicks can be arranged to form six congruent equilateral triangles.Rearrange the toothpicks to form sixteen congruent equilateral triangleseach of the same size as the original six.

PME558.83F.S85F.(R.I.Hess)

Let ABCD be a quadrilateral. Let each of the sides AB, BC, CD, DA bethe diagonal of a square. Let E, F . G. H be those vertices of the squaresthat lie outside the quadrilateral. Prove that EG and FH are perpendicular.



In a given circle the radii OA and OB are perpendicular. Let the circle onOB as diameter have center O′ and let O′A cut this new circle in point D.Then AD is the length of the side of a regular decagon inscribed in the givencircle. Also, let the tangent AQ to the new circle cut the given circle againat P . Then AP is the length of the side of a regular pentagon inscribed inthe given circle.

PME562.84S.S85S.(W.Blumberg)

Prove that tan 1 tan 61 = tan 3 tan 31.

PME565.84S.S85S.(W.Blumberg)

Let ABCD be a square and choose a point E on segment AB and pointF on segment BC such that angles AED and DEF are equal. Prove thatEF = AE + FC.


Find the exact value of sin 20 sin 40 sin 80.


(a) Find the largest regular tetrahedron that can be folded from a squarepiece of paper (with cutting).

(b) Prove whether it is possible to fold a regular tetrahedron from asquare piece of paper without overlapping or cutting.


Let ABCD be a parallelogram and construct directly similar triangles onsides AD, BC and diagonals AC and BD. See the figure, in which trianglesADE, ACH, BDF and BCG are the directly similar triangles. What re-strictions on the appended triangles are necessary for EFGH to be rhombus?


PME580.84F.S85F.(B.Prielipp)

Let a, b, c be the lengths of the sides of a triangle and s its semiperimeter.Prove that (

a

2

)a ( b2

)b ( c2

)c

≥ (s− a)a(s− b)b(s− c)c.

PME581.84F.S85F.(Rabinowitz)

If a triangle similar to a 3-4-5 triangle has its vertices at lattice points inthe plane, must its legs be parallel to the coordinate axes ?

PME582.84F.S85F.(W.Blumberg)

Suppose bc2 cosB = ca2 cosC = ab2 cosA. Prove that the triangle is equi-lateral.


Let ABC be any triangle with base BC. Let D be any point on side ABand E any point on side AC. Let PDE be an isosceles triangle with baseDE, oriented the same as ABC, and with apex angle P equal to angle A.Find the locus of all such points P .


Two circles are externally tangent and tangent to a line L at points A andB. A third circle is inscribed in the curvilinear triangle triangle boundedby these two circles and L and it touches L at point C. A fourth circle isinscribed in the curvilinear triangle bounded by line L and the circles atA and C and it touches the line at D. Find the relationship between thelengths AD, DC and CB.


Find the smallest n such that there exists a polyhedron of nonzero volumeand with n edges of lengths 1,2, . . . , n.


Given isosceles triangle ABC and a point O in the plane of the triangle,erect directly similar isosceles triangles POA, QOB and ROC (but not


necessarily similar to ABC). Prove that the apexes P , Q, R of thesetriangles determine a triangle similar to ABC.

PME604.85F.S86F.(D.Iny)

A unit square is covered by n congruent equilateral triangles of side s withoutthe triangles overlapping each other. Find the minimum value of s for n =1, 2, 3.


Triangles ABC and A′B′C ′ are right triangles with right angles at C andC ′. Prove that if s

r = s′r′ , then s

R < s′R′ .

PME616.86S.S87S.(D.P.Mavlo)

Prove that in any triangle

tan A2 + tan B

2 + tan C2

cot A2 + cot B

2 + cot C2

≤ 827

+(

tanA

2tan

B

2tan

C

2

)2

,


PME620.86S.(Garfunkel)

A triangle ABC isinscribed in an equilateral triangle PQR. The angle bisec-tors of triangle ABC are drawn and extended to meet the sides of trianglePQR in points A1, B1, C1. Now draw the angle bisectors of A1B1C1 tomeet the sides of triangle PQR at A2, B2, C2. Repeat the procedure. Proveor disprove that triangle AnBnCn tends to equilateral as n tends to infinity.(This result has been proved when a circle is used instead of triangle).


(i) Characterize all triangles whose angles and whose sides are both in arith-metic progression.

(ii) Characterize all triangles whose angles are in arithmetic progressionand whose sides are in geometric progression.


PME622.86S.(W.Blumberg)

Let P be the cneter of an equilateral trianlge ABC and let C be any circlecentered at P and lying entirely within the triangle. Let BR and CS betangents to the circle such that point R is closer to C than to A and S iscloser to A than to B. Prove that line RS bisects side BC.


Prove that

cosA cosB cosC ≤ (1 − cosA)(1 − cosB)(1 − cosC).

PME630.86F.S87F.(R.Euler)

Evaluatej∏

m=1

sinmπ

2j + 1.


Let ABC be a triange with ABC = ACB = 40. Let BD be the bisectorof ABC and produce it to E so that DE = AD. Find the measure of BEC.


The circle with center O is an excircle of triangle ABC. Then BK is drawnso that KBA = AOC, and OA is produced to meet BK in D. Provethat OCBD is a cyclic quadrilateral.

PME644.87S.S88S.(R.I.Hess)

In the figure below, prove that regions A and B have equal areas.


Let M be an arbitrary point on segment CD of trapezoid ABCD havingsides AD and BC parallel. Let S, S1, and S2 be the areas of triangles ABM ,BCM , and ADM . Prove that

S ≥ 2min(S1, S2).



In any triangle ABC, prove that

∏cos

A

2≤

√3

6

∑cos2 A

2.

PME653.87F.S88F.(R.C.Gebhardt and C.H.Singer)

A small square is constructed inside a square of area 1 by marking off seg-ments of length 1

n along each side as shown below. For n = 4, the side s ofthe small square is 1

5 . For what other values of n is s the reciprocal of aninteger ?


IN ABD, B = 120. There is a popint C on the side AD such that ABC = 90, AC = 3

√2, and BD = 2

AC . Find the lengths of AB and CD.


Let ABC be any triangle and extend side AB to A′, side BC to B′, and sideCA to C ′ so that B lies between A andA′ etc., and BA′ = λAB, AC ′ = λCA,and CB′ = λBC. Find the value of λ that the area of triangle A′B′C ′ is 4times the area of ABC.


Evaluatesin6 π

8+ sin6 3π

8+ sin6 5π

8+ sin6 7π

8.


(a) How close to a cubical box can you get if the sides and the diagonal ofa rectangular parallelepiped are all integral ?

(b) How close can you get to a cube if all the face diagonals must beintegral too ?


Let AB be an edge of a regular tessereact (a four-dimensional cube) and letC be the tesseract’s vertex that is furthest from A. Find the measure ofanlge ACB.


PME675.88S.S89S.(J.H.Scott)

Erect a semicircle on a segment AB as diameter. From point D on thesemicircle drop a perpendicular to point C on AB. Draw a circle tangentto CB at J and tangent to the semicircle and to segment CD. Prove thatangles CDJ and JDB have equal measures.


In any triangle ABC,

cosA cosB cosCcos A

2 cos B2 cos C

2

≤√

39.


(a) Given three concentric circles construct an isosceles right triangle so thatits vertices lie one on each circle.

(b) Is the construction always possible ?

PME684.88F.S89F.(D.P.Mavlo)

Erdos and Hans Debrunner, Elem. der Math. 11 (1956) p.20), proved thefollowing theorem: Let D, E, F be points on the interiros of sides BC,CA, AB of triangle ABC. Then the area of DEF cannot be less than thesmallest of the three other triangles formed:

[DEF ] ≥ min([AEF ], [CDE], [BFD]).

(a) Prove this generalization of the Erdos - Debrunner theorem: for somefixed real number α, if −∞ < t ≤ t, then

[DEF ] ≥(

[AEF ]t + [CDE]t + [BFD]t

3

) 1t

.

(b) Determine all the cases where equality holds.(c) Prove that for t = −1, the inequality of part (a) is equivalent to the

inequality

(1 + xyz)(

1x(y + z)

+1

y(z + x)+

1z(x+ y)

)≥ 3,

with equality if and only if x = y = z.



In any triangle ABC with C < 45, and given any other angle D with0 < D < 45, prove that

b cosD − c cos(A−D) < a.

PME690.88F.S89F.(D.Iny)

A unit square is covered by 5 circle of equal radius. Find the minimumnecessary radius. See also PME507.82F.



√sinA+

√sinB +

√sinC ≥ 6

√√

3 sinA

2sin

B

2sin

C

2.

PME697.89S.S90S.(K.Goggin)

Circle (B) is internally tangent to circle (A) at K and to diameter V AWat cneter A. Circle (C) is internally tangent to circle (A) at Z, externallytangent circle (B) at L. Find the ratios of the areas of the three circles toone another.


Let L and B be nonnegative numbers such that√

3L + 9B = 9√

3. Provethat in any triangle ABC,

tan A2 + tan B

2 + tan C2

cot A2 + cot B

2 + cot C2

≥ L

(tan

A

2tan

B

2tan

C

2

)2

,+B(

tanA

2tan

B

2tan

C

2

).



In right triangle ABC with right angle at C the altitude CD and the medianCE are drawn. Determine the ratio of the sides containing the right angleif AB = 3DE.


PME707.89F.S90F.(Klamkin)

From a point R taken on any circular arc PQ of less than a quadrant, twosegments are drawn, one to an extremity P of the arc and the other RSperpendicular to the chord PQ of the arc and terminated by it. Determinethe maximum of the sum PR+RS of the lengths of the two segments.

This problem without solution is given in Todhunter’s Trigonometry.


Find a Mascheroni construction (a construction using only compass, nostraightedge allowed) for the orthic triangle of an acute triangle.

PME709.89F.S90F.(N.Schaumberger)

In any triangle ABC,

a2b2 + b2c2 + c2a2 ≥ 122 +18s4.

PME711.89F(correction, 90F).S90F.(J.N.Boyd)

A pentagon is constructed with five segments of lengths 1,1,1,1, and w. Findw so that the pentagon will have the greatest area.


Which of the following triangle inequalities, if any, are valid ?(a) max(ha, hb, hc) ≥ min(ma,mb,mc);(b) max(wa, wb, wc) ≥ min(ma,mb,mc);(c) mid (wa, wb, wc) ≥ min(ma,mb,mc).



2 +∏

cosB − C

2≥ 2

∑cosA.


Given a non-obtuse triangle with altitude CD = h, drawn to side AB, denotethe inradii of triangles ACD BCD and ABC by r1, r2 and r3 respectively.Prove that if r1 + r2 + r3 = h, then the triangle is right angled at C.



(a) Show that on the lattice points in the plane one cannot have the verticesof an equilateral triangle.

(b) What about a tetrahedron in space ?


(a) What numbers cannot be a leg of a Pythagorean triangle ?(b) What numbers cannot be a hypotenuse of a Pythagorean triangle ?(c) What numbers can be neither a leg nor a hypotenuse of a Pythagorean

triangle ?


Construct squares outwardly on the sides of a triangle ABC. Prove ordisprove that the centers A′, B′, C ′ of these squares form a triangle thatis closer than being equilateral than is ABC. A proof would show that ifthe process were repeataed on triangle A′B′C ′, etc., that triangle AnBnCn

would approach equilateral as n approach infinity.


Let A and B be the ends of the diameter of a semicircle of radius r and letP be any point on the semicircle. Let I be the incenter of triangle APB.Find the locus of I as P moves along the semicircle.


Let ABC be inscribed in a circle. Draw a line through A to intersect sideBC at D and the circle again at E. Without resorting to calculus, provethat AD

DE is minimum when AD bisects angle A.


Let ABC be a triangle with inscribed circle (I) and let the line segmentsAI, BI, CI cut hte incircle at A′, B′, C ′ respectively. Prove that

sinA′ + sinB′ + sinC ′ ≥ cosA

2+ cos

B

2+ cos

C

2.


PME755.91S.S92S.(S.Rabinowitz)

In triangle ABC, a circle of radius p is inscrbed in the wedge bounded bysides AB and BC and the incircle (I) of the triangle. A circle of radius qis inscrbed in the wedge bounded by sides AC and BC and the incircle. Ifp = q, prove that AB = AC.

PME757.91S.S92S.(P.A.Coatney)

Find the overall height of the pyramid formed from four spherical balls ofradius r.

PME759.91F.S92F.(J.E.Wetzel)

Call a plane arc special if it has length 1 and lies on one side of a line throughits endpoints. Show that anhy special arc can be contained in an isoscelesright triangle of hypotenuse 1.

PME760.91F.S92F.(J.E.Wetzel)

Napoleon’s Theorem is concerned with erecting equilateral triangles out-wardly on the sides of a given triangle ABC. Then DEF is the triangleformed by the third vertices of these equilateral triangles BCD, CAE, andABF . Lemoine asked in 1868 if one can reconstruct ABC when onlyDEF is given. Shortly afterward, Keipert showed that the constructionis to erect outward equilateral triangles EFX, FDY and DEZ on triangleDEF , and then A, B, C are the midpoints of the segments DX, EY , FZ.His proof was quite tedious. Find a simple proof of Keipert’s construction.


Given triangle ABC, draw rays inwardly from each vertex to form a triangleA′B′C ′ such that B′, C ′, A′ lie on rays AA′, BB′, CC ′ respectively, and

BAB′ = ACA′ = CBC ′ = α,

as shown in the figure. Prove that(a) A′B′C ′ is similar to ABC;(b) the ratio of similitude is cosα − sinα cotω, where ω is the Brocard

angle of triangle ABC.


Solution. (a) Note that B′A′C ′ = A′AC + A′CA = A′AC + α = A′AC + A′AB = BAC. Similarly, C ′A′B′ = CAB. It follows thatthe triangles A′B′C ′ and ABC are similar.

(b) Applying the sine law to triangles AA′C ′, BCC ′ and ABC, we have

A′C =b sin(A− α)

sinA,

CC ′ =a sinαsinC

=b sinA sinαsinB sinC

.

It follows that

A′C ′ = b

[sin(A− α)

sinA− sinA sinα

sinB sinC

]

A′C ′

AC=

sin(A− α)sinA

− sin(C +B) sinαsinB sinC

= cosα− sinα[cotA+ cotB + cotC]= cosα− sinα cotω,

where ω is the Brocard angle of the triangle ABC and it is well known that

cotω = cotA+ cotB + cotC.


If ABC is a triangle in which c2 = 4ab cosA cosB, prove that the triangleis isosceles.


In a given circle (O) a chord CD is drawn to intersect diameter AOB atpoint E. Three circles are inscribed, the first two inthe sectors BEC andBED, ang the third in the opposite segment CED. Let the circle in sectorBEC touch CE at J and let the circle in sector BED touch DE at N . Seethe figure.

Suppose the three inscribed circles have equal radii.(a) Show that CD is perpendicular to AB.(b) Find the ratio AE

EB .(c) Find the ratio AD

AB .(d) Find the ratio CD

AB .(e) Show that the rectangle JKMN on JN as base and with opposite

side KM passing through A circumscribes the third inscribed circle.(f) Show that the rectangle JKLD and NMLD are golden rectangles.


PME780.92S,(correction 92F,93S).S94F.(R.S.Luthar)

Let ABCD be a parallelogram with A = 60. Let the circle through A,B, and D intersect AC at E. See the figure. Prove that BD2 +AB ·AD =AE · AC.


Erect squares ADEF , BKL and CDGH as shown in the figure, on thesegments AD, DC, and BD, where D is any point on side CA of giventriangle ABC. Let X, Y and Z be the centers of the erected squares. Provethat triangles ABC and XY Z are similar and the ratio of similarity is

√2.


Bottema 12.55: for a triangle ABC with an angle ≥ 120,

2(R1 +R2 +R3)2 ≥ (a2 + b2 + c2) + 4√

3,

where R1, R2, R3 are the respective distances from an arbitrary point Pinside the triangle to its sides. Item 12:55 further states that for a trianglein which A ≥ 120,

(R1 +R2 +R3)2 ≥ (b+ c)2.

Show that the first inequality is true for all triangles.


In any triangle ABC, ∑sin2A∑cos2 A

2

≥∏

sinA∏cos A

2

.

Solution. We begin by establishing the basic relations

a2 + b2 + c2 = 2s2 − 2(4R + r)r, (1)ab+ bc+ ca = s2 + (4R+ r)r, (2)abc = 4Rrs. (3)

Since R = abc4 and r =

s , (3) is immediate.Also,

r2s2 = 2 = s(s− a)(s − b)(s− c)


= −s4 + s2∑

bc− abcs

= −s4 + s2∑

bc− 4Rrs2,

from which we obtain (2). (1) follows from∑a2 = (

∑a)2 − 2

∑bc.

Since∑a2 ≥∑

bc with equality if and only if a = b = c, we have,

s2 ≥ 3(4R + r)r, (4)

with equality if and only if the triangle is equilateral. Now, since sinA = a2R ,

and

cosA

2=

√s(s− a)bc

, sinA

2=

√(s− b)(s − c)

bc,

we have ∑sin2A∑cos2 A

2

=1

4R2

∑a2

sabc

∑a(s− a)

=1

4R2

∑a2

sabc

∑a(s− a)

=abc

4R2s·

∑a2

2s2 −∑ a2=

r

R

(s2

(4R + r)r− 1

),

using (1) and (2). On the other hand,∏

sinA∏cos A

2

= 8∏

sinA

2=

8(s− a)(s − b)(s− c)abc

=8(−s3 + s

∑bc− abc)

abc=

2(−s2 +∑bc− 4Rr)

Rr=

2rR.

The inequality in question is equivalent to s2

(4R+r)r −1 ≥ 2. This is equiv-alent to (4) above. Equality holds if and only if the triangle is equilateral.

PME793.92F.S93S.(D.Bennewitz)

Given any trapezoid, its diagonals divide its interior area into four triangularareas: A and B adjacent to the parallel bases, and C and D adjacent to thenonparallel sides.

(a) Prove that the areas C and D are equal and that A ·B = C ·D.(b) Find area C in terms of the lengths of the altitude and the bases of

the trapezoid.



In any triangle, prove that

∑√tan

A

2<

√3∑√

cscA.


Show that4 arctan

1 − x

1 + x= π − 4 arctan x.

PME807.93S.S94S.(F.Smarandache)

In terms of the lengths a, b, c of a given triangle ABC, find the length ofthe segment PQ of the normal to the side BC at its midpoint M cut off bythe other two sides.

PME808.93S.S94S.(S.H.Brown)

A circle (R) is inscribed in the unit square ABCD in the unit square ABCDand touches the sides of the square at S, T , U and V , as shown in theaccompanying figure. Another circle (r) is inscribed in the region ASVoutside circle (R) and inside the square at vertex A.

(a) Find the area of the shaded region inside region ASV and outsidecircle (r).

(b) If the sequence of smaller circles is continued indefinitely, each suc-cessive circle inscribed between the preceding and the corner A of the square,find the limit of the shaded region.

PME809.93S.S94S.(D.Iny)

In triangle ABC let AD and BE be any two cevians intersecting at a pointF . Find the ratios BD

DC and AFFD in terms of the ratios AE

EC and BFFE .

PME811.93F.S94F.(T.Moore)

A primitive Pythagorean triple (a, b, c) is prime if both a and c are prime.(a) If (a, b, c) is prime deduce that b = c− 1.(b) Find all prime, primitive Pythagorean triples in which a and c are(i) twin primes;


(ii) both Mersenne primes;(iii) both Fermat primes;(iv) one a Mersenne, the other a Fermat prime.

PME812.93F.S94F.(G.P.Evanovich)

If n ≥ 2 is a positive integer, prove that

n∑j=1

cos2jπn

=n∑

j=1

sin2jπn

= 0.


Given a triangle ABC with sides a, b, c, and a triangle A′B′C ′ with sides12 (b+ c), 1

2(c+ a), 12(a+ b). Prove that r′ ≥ r.

PME817.93F.S94F.(A.Cusumano)

In the accompanying figure, squares CEHA and AIDB are erected exter-nally on sides CA and AB of triangle ABC. Let BH meet IC at O and ACat G, ang let CI meet AB at F .

(a) Prove that points D, O, and E are collinear.(b) Prove that angles HOE, EOC, AOH, and AOI are each 45.(c) If ACB is a right angle, then prove that E, F , and G are collinear.Find an elegant proof for parts (a) and (b), both of which are known to

be true whether the squares are rected both externally or both internally (seeAMME831.49p.406–407). Part (c) is a delightful result that also should beknown, but appears to be more difficult to prove. (See also PME895.96F.)

PME825.94S.S95S,95F.(Bankoff)

Let O be a point inside the equilateral triangle ABC whose side is of lengths. Let OA, OB, OC have lengths a, b, c respectively. Given the lengths a,b, c, find length s.

See also AMM 3904.392.S411, CMJ187.812.S824, and Bottema, On thedistance of a point to the vertices of a triangle, Crux Math. 10 (1984) pp.242– 246.



Let P be a point on diagonal BD of square ABCD and let Q be a point onside CD such that APQ is a right angle. Prove that AP = PQ. Solution.Let the perpendicular to AB through P meet AB and CD respectivelyat H and K. Clearly, in the right triangles APH and PKQ, APH =π2 − QPK = PQK. Furthermore, PK = DK = AH. It follows thatthe right triangles APH and PKQ are congruent, and their hypotenusesAP and PQ are equal in length. Second solution. Note that APQD is

a cyclic quadrilateral since it has two opposite right angles. It follows that PAQ = PDQ = 45, and in the right triangle APQ, AP = PQ.


Let T and T ′ denote two triangles with respective sides (a, b, c) and (a′, b′, c′),where

a′2 = s(s− a), b′2 = s(s− b), c′2 = s(s− c).

Prove that(i) s ≥ s′;(ii) R ≥ R′;(iii) r′ ≥ r;(iv) ′

s′2 ≥ s2 .

PME846.94F.S95F.(M.A.Khan)

Let N , L, M be points on sides AB, BC, CA of a given triangle ABC suchthat

0 <AN

AB=BL

BC=CM

CA= k < 1.

Let AL meet CN at P and BM at Q, and let BM and CN meet at R. Drawlines parallel to CN through A, parallel to AL through B, and parallel toBM through C. Let XY Z be the triangle formed by these three new lines.Prove that

(a) triangles ABC, PQR and XY Z have a common centroid;(v) if the areas of PQR, ABC and XY Z are in geometric progression,

then k =√

3 − 1.

PME847.94F.S95F(D.P.Mavlo)

The midline of an isosceles trapezoid has length L and its acute angle is α.Determine the trapezoid’s area. Answer. L2 sinα.


PME851.95S.S96S. (B.Correll)

In ABC let cevian AD bisect side BC and let cevians BE and BF trisectside CA. Let AD intersect BE at P and BF at R, and let CP meet BF atQ, If the area of ABC is 1, find the area of triangle PQR.

PME852.95S.S96S. (R.H.Wu)

Let E be a point inside square ABCD with BE = x, DE = y, and CE = z.If x2 + y2 = 2z2, find the area of ABCD in terms of x, y,z.

PME856.95S.S96S. (P.S.Bruckman)

Starting with a regular n−gon whose side is of unit length, snip off congruentisosceles triangle from each of its vertices, resulting in a regular 2n−gon.Repeat the process indefinitely. Find the ratio of the area of the limitingcircle to that of the original n−gon. Solution. All these regular polygonshave the same inscribed circle. The ratio is therefore

π

n tan πn

=πn

tan πn

.

PME865.95F.S96F.(M.A.Covas)

Let ABC be a triangle with sides of lengths a, b, c, semiperimeter s and areaK. Show that, if

∑a(s − a) = 4K, then the three circles centered at the

vertices A, B, C and of radii s− a, s− b, s− c respectively, are all tangentto thesame straight line.

PME869.95F.S96F.(R.Behboudi)

Consider an ellipse with center O and major and minor axes AB and CDrespectively. Let E and F be points on segment OB so that OE2 +OF 2 =OB2.At E and F erect perpendiculars to cut arc BC at G and H respec-tively. Show that the areas of sectors OBH and OGC are equal.

PME871.95F.S96F.(M.A.Covas)

Let ABCD be an isosceles trapezoid with major base BC. If the altitudeAH is the mean proportional between the bases, then show that each sideis the arithmetic mean of the bases, and show that the projection AP of thealtitude on side AB is the harmonic mean of the bases.


PME881.96S.S97S;97F. (A.Cusumano)

Let ABC be an equilateral triangle with center D. Let α be an arbitrarypositive angle less than 30. Let BD meet CA at F . Let G be that pointon segment CD such that angle CBG = α, and let E be that point on FGsuch that FCE = α. Prove that DE is parallel to BC.

PME895.96F.S97F. (A.Cusumano)

Let ABC be an isosceles right triangle with right angle at C. Erect squaresACEH and ABDI outwardly on side AC and hypotenuse AB. Let CI meetBH at K, and let AO meet BC at J . Let DE cut AB at F and AC at G.It is known (PME817.94F) that DE passes through O. Let JF meet AHat S and let JG meet BH at T . Finally, let BH and AC meet at M andlet JM and CI meet at L.

(a) Prove that

1. ST is parallel to DOE,

2. JK is parallel to AC,

3. JG is parallel to AB,

4. AI passes through T ,

5. JF passes through I,

6. EK passes through M , and

7. BL pases through G.

(b) Which of these results generalize to an arbitrary triangle?

PME900.96F.S97F. (H.Eves)

Given the lengths of two sides of a triangle and that the medians to those twosides are perpendicular to each other, construct the triangle with euclideantools.

PME910.97S. (W. Chau)

A triangle whose sides have lengths a, b, c has area 1. Find the line segmentof minimum length that joins two sides and separates the interior of thetriangle into two parts of area α and 1 − α, where α is a given numberbetween 0 and 1.


PME910.97S.S98S. (N.Schaumberger)

If a, b, and c are the lengths of the sides of a triangle with semiperimeter sand area K, show that

(s

s− a

)a/(s−a)

+(

s

s− b

)b/(s−b)

+(

s

s− c

)c/(s−c)

≥ s4

K2.

PME919.97F.S98F. (C.W.Dodge)

Erect directly similar nondegenerate triangles DBC, ECA, FAB on sidesBC, CA, AB of triangle ABC. At D, E, F center circles of radii k · BC,k · CA, k ·AB respectively for fixed positive k. Let P be the radical centerof the three circles. If P lies on the Euler line of the triangle, show that italways falls on the same special point.

PME924.97F.S98F. (G.Tsapakidis)

Find an interior point of a triangle so that its projections on the sides of thetriangle are the vertices of an equilateral triangle.

Comment: The solution given by W.H.Peirce can actually be adaptedto give a simple description of the points. Peirce solved the problem bycalculating the barycentric coordinates of the point P . Homogenizing, weobtain

a sin(α± 60) : b sin(β ± 60) : c sin(γ ± 60)

These points divide the segment OK harmonically, in the ratio a2 + b2 +c2 : 4

√3.

They are the isogonal conjugates of the points

a

sin(α± 60):

b

sin(β ± 60):

c

sin(γ ± 60),

which are the isogonal centers of the triangle.

PME936.98S. (J.Garfunkel)

Given the Malfatti configuration, where three mutually external, mutuallytangent circles with centers A′, B′, C ′ are inscribed in a triangle ABC so


that circle (A′) is tangent to the two sides of angle A, circle (B′) is tangentto the sides of angle B, and (C ′) to the sides of C. If A ≤ B ≤ C, and A < C, then prove that we have C ′ − A′ < C − A.

PME937.98S. (R.S.Luthar)

Let I be the incenter of triangle ABC, let AI cut the triangle’s circumcircleagain at point D, and let F be the foot of the perpendicular dropped fromD to side BC, as shown in the figure. Prove that DI2 = 2R ·DF , where Ris the circumradius of triangle ABC.

PME938.98S. (R.S.Luthar)

Find the locus of the midpoints M of the line segment in the first quadrantlying between the two axes and tangent to the unit circle centered at theorigin.

PME939.98S. (Khiem Viet Ngo)

In the accompanying figure both quadrilaterals ABCD and MNPQ aresquares, each side of square ABC has length 1, and the five inscribed circleare all congruent to one another. Find their common radius.

PME945.98F. (J.Garfunkel)

Let A, B, C be the angles of a triangle and A′, B′, C ′ those of anothertriangle with A ≥ B ≥ C, A > C, A′ ≥ B′ ≥ C ′, and A′ > C ′. Prove ordisprove that if A− C ≥ 3(A′ − C ′), then

∑cos

A

2≤∑

sinA′.

PME946.98F. (Ayoub B. Ayoub)

Let M be a point inside (outside) triangle ABC if A is acute (obtuse) andlet MBA+ MCA = 90.

(a) Prove that (BC ·AM)2 = (AB · CM)2 + (CA · BM)2,(b) Show that the Pythagorean theorem is a special case of the formula

of part (a).


PME950.98F. (S.B.Karmakar)

Let c > b > a > 0 be the lengths of the sides of an obtuse triangle; let m bea prime and n an evern positive integer such that 1 < d = m

n < 2. Withoutusing Fermat’s last theorem proe that the equation

ad + bd = cd

cannot be satisfied if a, b, and c are relatively prime in pairs.

PME952.98F. (P.A.Lindstrom)

Let A, B, C denote the measures of the angles and a, b, c the opposite sidesof a triangle. Show that

sinA sinB+sinB sinC+sinC sinA =(a+ b+ c)(b+ c− a)(c+ a− b)(a+ b− c)(bc + ca+ ab)

4a2b2c2.


Crux Mathematicorum

Geometry Problems (1975 – 2000)

Crux 5.S1.15.(L.Sauve)

Prove that, if (a, b, c) and (a′, b′, c′) are primitive Pythagorean triples, witha > b > c and a′ > b′ > c′, then either

aa′ ± (bc′ − cb′) or aa′ ± (bb′ − cc′)

are perfect squares.

Crux 14.S1.28. (V.Linis)

If a, b, c are the lengths of three segments which can form a triangle, showthe same for 1

a+c ,1

b+c ,1

a+b .

Crux 15.S1.28. (H.G.Dworkschak)

Let A, B, C be three distinct points on a rectangular hyperbola. Prove thatthe orthocenter of ABC lies on the hyperbola.

[Solution by Leo Sauve]: If A, B, C are the points (a, ka), (b, k

b ), (c, kc )

on the hyperbola xy = k, the orthocenter is the point (t, kt ), with t = − k2

abc .

Crux 18.S1.31;2.42,69. (J.Marion)

Montrer que, dans un triangle rectangle dont les cote ont 3, 4 et 5 unites delongueur, aucun des angles aigus n’est un multiple rationnel de π.


How many different triangles can be formed from n straight rods of lengths1,2, . . . , n ?

See also CMJ503.933.



A paper triangle has base 6 cm and height 2 cm. Show that by three orfewer cuts the sides can cover a cube of edge 1 cm.

Crux 27.S1.44. (L.Sauve)

Soient A, B, et C les angles d’un triangle. Il est facile de verifer que siA = B = 45, alors

cosA cosB + sinA sinB sinC = 1.

La proposition reciproque est-elle vraie ?


Cut a square into a minimal number of triangles with all angles acute.


Construct a square given a vertex and a midpoint of one side.


On the sides CA and CB of an isosceles right triangle ABC, points D andE are chosen such that CD = CE. The perpendiculars from D and Con AE intersect the hypotenuse AB in K and L respectively. Prove thatKL = LB.

Crux 37.S1.62*. (M.Poirier)

E, F , G, and H are the midpoints of the sides AB, BC, CD and DArespectively of the convex quadrilateral ABCD. EX, FY , GZ and HTare drawn externally perpendicular to AB, BC, CD and DA, respectively,and EX = 1

2AB, FY = 12BC, GZ = 1

2CD, and HT = 12DA. Prove that

XZ = Y T and XZ ⊥ Y T .

[Restatement]: If squares on constructed externally on the sides of aconvex quadrilateral, the centers of the squares form a quadrilateral whosediagonals are equal and perpendicular to each other.

[Editor’s Comment]: This theorem is due to H. van Aubel, who was aprofessor in the athenee of Antwerp around 1880. It appears as Proble 10 on


p.23 of Coxeter’s Introduction to Geometry, and a solution (different fromthose given in Crux) is given at the back of the book. Paul J. Kelly [vonAubel’s quadrilateral theorem, Math. Mag. 39 (1966) 35] generalized it tofour non-coplanar points.

Crux 38.S1.63*. (L.Sauve)

Consider the two triangles ABC and PQR show below. In ABC, ADB = BDC = CDA = 120. Prove that X = u+ v + w.

Crux 39.S1.64*;2.7. (M.Poirier)

On donne un point P a l’interieur dun triangle equilateral ABC tel que leslongueurs des segments PA, PB, PC sont 3,4, et 5 unites respectivment.Calculer laire du ABC.


Find the area of quadrilateral as a function of its four sides, given that thesums of opposite angles are equal.


Construct a square ABCD given its center and any two points M and Non its two sides BC and CD respectively.

Crux 46.S1.75. (F.G.B.Maskell)

1ha

+1hb

+1hb

=1r.


Solution. 2 = aha = bhb = chc = (a+ b+ c)r.


What is the area of a triangle in terms of its medians ?


Prove that if two circles touch externally, their common tangent is a meanproportion between their diameters.


From the centers of each of two nonintersecting circles tangents are drawnto the other circle. Prove that the chords PQ and RS are equal in length.(I have been told that this problem originated with Newton, but have notbeen able to find the exact reference).


Show that in any convex 2n−gon there is a diagonal which is not parallelto any of its sides.


Show that for any 13−gon there exists a straight line containing only one ofits sides. Show also that for every n > 13 there exists an n−gon for whichthe above statement does not hold.



Is there a polyhedron with exactly ten pentagons as faces ?


Prove that if the sides a, b, c of a triangle satisfy a2 + b2 = kc2, then k > 12 .

Crux 75.S2.10*. (R.D.Butterill)

M is the midpoint of chord AB of the circle with center C shown in thefigure below. Prove that RS > MN .

See also Crux 110.

[Solution:] MNRS = cosNPS.


Find all rational Pythagoras triples (a, b, c) such that

a2 + b2 = c2, and a+ b = c2.

Crux 89.S2.33. (V.Bradley and C.Robsertson)

A goat is tethered to a point on the circumference of a circular field of radiusr by a rope of length . For what value of will it be able to graze overexactly half of the field ?

Crux 93.S2.45,111. (H.G.Dworkschak)

Is there a convex polyhedron having exactly seven edges ?See also Crux 121.



If, in a tetrahedron, two pairs of opposite edges are orthogonal, is the thirdpair of opposite necessarily orthogonal ?


By euclidean methods divide a 13 angle into 13 equal parts.

Crux 102.S2.73. (L.Sauve)

Si, dans un ABC, on a = 4, b = 5, et c = 6, montrer que C = 2A.

Crux 103.S2.74. (Dworschak)

If cos αcos β + sinα

sinβ = −1, prove that

cos3 βcosα

+sin3 β

sinα= 1.

[Solution by Leo Sauve]: The following are equivalent:

1. cos αcos β + sin α

sin β = −1;

2. cos3 βcos α + sin3 β

sinα = 1;

3. sin(α+ β) = −12 sin 2β.

This last item is equivalent to

sin(β + β) + sin(β + α) + sin(α+ β) = 0.

By Crux 132, this means that the normal to the ellipse

x2

a2+y2

b2= 1

at points with eccentric angles α, β, β are concurrent. This is the case ifand only if the center of curvature for β, namely,(

c2

acos3 β, −c

2

bsin3 β

)

lies on the normal at α. . . .



Prove that, for any quadrilateral with sides a, b, c, d,

a2 + b2 + c2 >13d2.

Crux 107.S2.79. (V. Linis)

For which integers m and n is the ratio 4m2m+2n−mn an integer ?

This problem has a geometric application.

Crux 109.S2.81. (L. Sauve)

(a) Prove that rational points are dense on any circle with rational centerand rational radius.

(b) Prove that if the radius is rational the circle may have infinitely manyrational points.

(c) Prove that if even one coordinate of the center is irrational, the circlehas at most two ratinal points.

Crux 110.S2.84. (Dworschak)

(a) Let AB and PR be two chords of a circle intersecting at Q. If A, B,and P are kept fixed, characterize geometrically the position of R for whichthe length of QR is maximal.

(b) Give a euclidean construction for the point R which maximizes thelength of QR, or show that no such construction is possible.

[Solution.] (a) QR bisected by the diameter perpendicular to AB.(b) Not constructible in general.



Si u = (b, c, a) et v = (c, a, b) sont deux vecteurs non nuls dans l’espaceeuclidien reel a trois dimensions, quelle est la valeur maximale de l’angle(u, v) entre u et v ? Quand cette valeur maximale est-elle atteinte ?

Crux 115.S2.98,111,137*. (V. Linis)

Prove the following inequality of Huygens:

2 sinα+ tanα ≥ 3α, 0 ≤ α <π

2.

See also Crux 167.

Crux 119.S2.102. (J.A.Tierney)

A line through the first quadratnt point (a, b) forms a right triangle withthe positive coordinate axes. Find analytically the minimum perimeter ofthe triangle.

Crux 120.S2.102,139. (J.A.Tierney)

Given a point P inside an arbitrary angle, give a euclidean construction ofthe line through P that determines with the sides of the angle a triangle

(a) of minimum area;(b) of minimum perimeter.

Crux 121.S2.121,139. (L. Sauve)

For which n is there a convex polyhedron having exactly n edges ?See also Crux 93.

Crux 125.S2.120.* (B. Vanbrugghe)

A l’aide d’un compas seulement, determiner le centre inconnu d’un cercledonne.


Show that, for any triangle ABC,

|OA|2 sinA+ |OB|2 sinB + |OC|2 sinC = 2.


Crux 127.S2.124,140,221. (V. Linis)

A, B, C, D are four distinct points on a line. Construct a square by drawingtwo pairs of parallel lines through the four points.

Crux 132.S2.142,172;3.11. (L. Sauve)

If cos θ = 0, and sin θ = 0 for θ = α, β, γ, prove that the normals to theellipse

x2

a2+y2

b2= 1

at the ponts of eccentric angles α, β, γ are concurrent if and only if

sin(α+ β) + sin(β + γ) + sin(γ + α) = 0.

Crux 134.S2.151,173,222;3.12,44. (K.S.Williams)

ABC is an isosceles triangle with ABC = ACB = 80. P is the point onAB such that PCB = 70. Q is the point on AC such that QBC = 60.Find PQA.

See also Crux 175.

Crux 136.S2.153. (S.R.Conrad)

In ABC, C ′ is on AB such that AC ′ : C ′B = 1 : 2 and B′ is on AC suchthat AB′ : B′C = 4 : 3. Let P be the intersection of BB′ and CC ′, and letA′ be the intersection of BC and ray AP . Find AP : PA′.


On a rectangular billiard table ABCD, where AB = a and BC = b, oneball is at a distance p from AB and at a distance q from BC, and anotherball is at the center of the table. Under what angle α (from AB) must thefirst ball be hit so that after the rebounds from AD, DC, CB it will hit theother ball ?

Crux 139.S2.158. (D.Pedoe)

ABCD is a parallelogram, and a circle γ touches AB and BC and intersectsAC in the points E and F . Then there exists a circle δ which passes throughE and F and touches AD and DC.

Prove this theorem without using Rennie’s lemma. See Crux 2. p.65.


Crux 140.S3.13,46. (D.Pedoe)

(The Veness Problem) A paper cone is cut along a generator and unfoldedinto a plane sheet of paper. what curve in the plane do the originally planesections of the cone become ?

Crux 141.S2.174. (Bankoff)

What is wrong with the following proof of the Steiner – Lehmus theorem ?At the midpoints of the angle bisectors, I erect two perpendiculars which

meet in O; with O as center and AO as radius, I describe a circle which willevidently pass through the points A, M , N , C.

Now, the angles MAN , MCN are equal since the measure of each is arcMN

2 ; hence BAC = ACB, and triangle ABC is isosceles.


In a triangle ABC, the medians AM and BN intersect at G. If the radii ofthe inscribed circles in triangle ANG and BMG are equal, show that ABCis an isosceles triangle.


In square ABCD, AC and BD meet at E. Point F is in CD and CAF = FAD. If AF meets ED at G and if EG = 24, find CF .


In ABC, C = 60, and A is greater than B. The bisector of C meetsAB in E. If CE is a mean proportional between AE and EB, find B.

Crux 155.S2.198;3.22. (S.R.Conrad and I.Ewen)

A plane is tessellated by regular hexagons when the plane is the union ofcongruent regular hexagonal closed regions which have disjoint interiors. Alattice point of this tessellaton is any vertex of any of the hexagons.

Prove that no four lattice points of a regular hexagonal tessellation of aplane can be the vertices of a regular 4−gon.


Crux 158.S2.201. (A.Bourbeau)

Devise a euclidean construction to divide a given line segment into two partssuch that the sum of the squares on the whole segment and on one of itsparts is equal to twice the square on the other part. Solution. Let AB bea given segment. Extend AB to a point B′ so that BB′ = AB. Constructequilateral triangle AB′C. On AB′ mark a point P such that B′P = BC.Then the sum of the squares on AB and AP is twice the square on PB.

Remark. This is the same as Trigg’s solution. How would Euclid havejustifed such a construction?

Crux 165.S2.230. (D. Eustice)

Prove that, for each choice of n points in the plane (at least two distinct),there exists a point on the unit circle such that the product of the distancesfrom the point to the chosen points is greater than one.

See also Crux 173.


The first half of the Snellius - Huygens double inequality

13(2 sinα+ tanα) > α >

3 sinα2 + cosα

, 0 < α <π

2,

was proved in Crux 115. Prove the second half in a way that could havebeen understood before the invention of calculus.

See also Crux 115.

Crux 168.S2.233. (Garfunkel)

Let ta, tb, tc be the lengths of the bisectors of a triangle, and Ta, Tb, Tc theseangle bisectors extended until they are chords of the circumcircle. Prove that

abc =√tatbtcTaTbTc.

Crux 171.S3.26. (D.Sokolowsky)

Let P1 and P2 denote, respectively, the perimeters of ABE and ACDas shown. Without using circles, prove that

P1 = P2 ⇒ AB +BF = AD +DF.


See also Crux 2, p.108.

Crux 173.S3.47,68. (D. Eustice)

For each choice of n points on the unit circle (n ≥ 2), there exists a pointon the unit circle such that the product of the distances to th chosen pointsis greater than 2. Moreover, the product is 2 if and only if the n points arethe vertices of a regular polygon.

See also Crux 165.

Crux 175.S3.49. (A. Dunkels)

Consider the isosceles triangle ABC with vertical angle A = 20. On AC,one of the equal sides, a point D is marked off so that |AD| = |BC| = b.Find the measure of ABD.

See also Crux 134.

Crux 177.S3.50*,132. (K.S.Williams)

P is a point on th diameter AB of a circle whose center is C. On AP , BPas diameters, circles are drawn. Q is the center of a circle which touchesthese three circles. What is the locus of Q as P varies ?

Crux 180.S3.50. (K.S.Williams)

Through O, the midpoint of a chord AB of an ellipse, is drawn any chordPOQ. The tangents to the ellipse at P and Q meet AB at S and T respec-tively. Prove that AS = BT .


Crux 181.S3.51. (Trigg)

A polyhedron has one square face, two equaliteral triangular faces attachedto opposite sides of the square, and two isosceles trapezoidal faces, each withone edge equal to twice a side of the square. What is the volume of thispentahedron in terms of a side of the square ?

Crux 189.S3.74,193,252. (K.S.Williams)

If a quadrilateral circumscribes an ellipse, prove that the line through themidpoints of its diagonals passes through the center of the ellipse.

Crux 192.S3.79*. (Honsberger)

Let D, E, F denote the feet of the altitudes of ABC, and let (X1,X2),(Y1, Y2), (Z1, Z2) denote the feet of perpendiculars from D, E, F respec-tively, upon the other two sides of the triangle. Prove that the 6 points X1,X2, Y1, Y2, Z1, Z2 lie on a circle.

Crux 199.S3.112,298. (Dworschak)

If a quadrilateral is circumscribed about a circle, prove that its diagonalsand the two chords joining the points of contact of opposite sides are allconcurrent.

Crux 200.S3.134,228. (L.Sauve)

(a) Prove that there exist triangles which cannot be dissected into two orthree isosceles triangles.

(b) Prove or disprove that, for n ≥ 4, every triangle can be dissectedinto n isosceles triangles.


A circle intersects the sides BC, CA and AB of triangle ABC in the pairsof points X, X ′, Y , Y ′, and Z, Z ′ respectively. If the perpendiculars at X,Y and Z to the respective sides BC, CA and AB are concurrent at a pointP , prove that the respective perpendiculars at X ′, Y ′ and Z ′ to the sidesBC, CA and AB are concurrent at a point P ′.


Crux 210.S3.160,196. (Klamkin)

P , Q, R denote points on the sides BC, CA and AB respectively of a giventriangle ABC. Determine all triangles ABC such that if

BP

BC=CQ

CA=AR

AB= k(= 0,

12, 1),

then PQR (in some order) is similar to ABC.

Crux 213.S3.160. (W.J.Blundon)

(a) Prove that the sides of a triangles are in arithmetric progression if andonly if s2 = 18Rr − 9r2.

(b) Find the correpsonding result for geometric progression.

Crux 218.S3.172. (G.W.Kessler)

Everyone knows that the altitude to the hypotenuse of a right triangle is themean proportional between the segments of the hypotenuse. The median tothe hypotenuse also has this property. Does any other segment from vertexto hypotenuse have the property ?


C is a point on the diameter AB of a circle. A chord through C, perpen-dicular to AB, meets the circle at D. A chord through B meets CD at Tand arc AD at U . Prove that there is a circle tangent to CD at T and toarc AD at U .

Crux 222.S3.200. (B.McColl)

Prove thattan

π

11tan

2π11

tan3π11

tan4π11

tan5π11

=√

11.


Without using any table which lists Pythagorean triples, find the small-est integer which can represent the area of two noncongruent primitivePythagorean triangles.


Crux 224.S3.203. (Klamkin)

Let P be an interior point of a given n−dimensional simplex of vertices A1,A2, . . . , An+1. Let Pi, (i = 1, 2, . . . , n + 1) denote points on AiP such thatAiPiPiP

= 1ni

. Finally, let Vi denote the volume of the simplex cut off fromthe given simplex by a hyperplane through Pi parallel to the face of thegiven simplex opposite Ai. Determine the minimum value of

∑Vi and the

location of the corresponding point P .


C is a point on the diameter AB of a circle. A chord through C, perpen-dicular to AB, meets the circle at D. Two chords through B meets CD atT1, T2, and arc AD at U1, . U2 respectively. It is known from Problem 220that there are circles C1, C2 tangent to CD at T1, T2 and to arc AD at U1,U2 respectively. Prove that the radical axis of C1 and C2 passes through B.

Crux 229.S3.231. (K.M.Wilke)

On an examinantion, one question asked for the largest angle of the trianglewith sides 21, 41, 50. A student obtained the correct answer as follows: LetC denote the desired angle; then

sinC =5051

= 1 +941.

But sin 90 = 1 and 941 = sin 1240′49′′. Thus,

C = 90 + 1240′49′′ = 10240′49′′,

which is correct. Find the triangle of least area having integer sides andpossessing this property.

Crux 232.S3.238;4.17. (V.Linis)

Given are five points A, B, C, D, E in the plane, together with the seg-ments joining all pairs of distinct points. The areas of the five trianglesBCD, EAB, ABC, CDE, DEA being known, find the area of the pen-tagon ABCDE.

The above problem with a solution by Gauss was reported by Schu-macher. The problem was given by Mobius in his book on the Observatoryof Leipzig, and Gauss wrote his solution in the margins of the book.



The three points (1), (2), (3) lie in this order on an axis, and the distances[1, 2] = a, and [2, 3] = b are given. Points (4) and (5) lie on one side of theaxis, and the distance [4, 5] = 2c > 0 and the angles (415) = v1, (425) = v2,(435) = v3 are also known. Determine the position of the points (1), (2),(3) relative to (4) and (5).

Gauss gave a solution to this problem which was found in a boook onnavigation [Handbuch der Schiffahrtskunde von C. Rumker, 1850, p.76].

Crux 234.(correction 3.154).S3.257. (V.Linis)

If sin 2nπ13 = ± sin π

13 , prove that

cosπ

13cos

2π13

cos4π13

· · · cos 2n−1π

13= ± 1

2n.

Gauss’ remark: Inspect a polygon !

Crux 242.S3.266*. (B.McColl)

Give a geometrical construction for determining the focus of a parabolawhen two tangents and their points of contact are given.

Crux 244.S4.19*. (S.R.Conrad)

Solve the following problem, which can be found in Integrated Algebra andTrigonometry, by Fisher and Ziebur, Prentice Hall, (1957) p. 259:

A rectangular strip of carpet 3 ft. wide is laid diagonally across the floorof a room 9 ft. by 12 ft. so that each of the four corners of the strip touchesa wall. How long is the strip ?

Crux 245.S4.21. (Trigg)

Find the volume of a regular tetrahedron in terms of its bimedian b. (Abimedian is a segment joining the midpoints of opposite edges).

Crux 248.(correction 3.154).S4.26 (D.Sokolowsky)

Circle (Q) is tangent to circles (O), (M), N) as shown in the figure, andFG is the diameter of (Q) parallel to diameter AB of (O). W is the radicalcenter of circles (M), (N), (Q). Prove that WQ is equal to the circumradiusof PFG.


Crux 255.S4.52. (B.Hornstein)

In the adjoining figure, the measures of certain angles are given. Calculatex in terms of α, β, γ, δ.

Crux 256.S4.53,102. (H.L.Nelson)

Prove that an equilateral triangle can be dissected into five isosceles trian-gles, n of which are equilateral, if and only if 0 ≤ n ≤ 2.

Crux 257.S4.54*. (W.A.McWorter)

Can one draw a line joining two distant points with a BankAmericard ?

Crux 260.S4.58;8.80. (W.J.Blundon)

Given any triangle (other than equilateral), let P represent the projectionof the incenter I on the Euler line OGNH. Prove that P lies between Gand H. In particular, prove that P coincides with N if and only if one angleof the given triangle is 60.

Crux 268.S4.78*. (G.Salvatore)

Show that in ABC with a ≥ b ≥ c, the sides are in arithmetic progressionif and only if

2 cotB

2= 3(tan

C

2+ tan

A

2).


Crux 270.S4.82*. (D.Sokolowsky)

Call a chord of a triangle a segment with enpoints on the sides. Show thatfor very acute angled triangle there is a unique point P through which passthree equal chords each of which is bisected by P .

See also editor’s comment on Crux 624.S8.111.

Crux 271.S4.84. (S.Avital)

Find all possible triangle ABC which have the property that one can drawa line AD, outside the triangular region, on the same side of AC as AB,which meets CB extended in D so that triangles ABD and ACD will beisosceles.

Crux 275.S4.105. (G.W.Kessler)

Given are the points P (a, b) and Q(c, d), where a, b, c, d are all rational. Finda formula for the number of lattice points on the segment PQ.

Crux 278.S4.110. (W.A.McWorter)

If each of the medians of a triangle is extended beyond the sides of thetriangle to 4

3 its length, show that the three new points formed and thevertices of the triangle all lie on an ellipse.


On donne sur une droite trois points distincts A, O, B tels que O est entreA et B, et AO = OB. Montrer que les trois coniques ayant deux foyers etun sommet aux trois points donnes sont concourantes en deux points.


Given a sector AOD of a circle (see figure), can a straightedge and compassconstruct a line OB so that AB = AC ?


Crux 288.S4.136*. (W.J.Blundon)

Show how to construct a triangle given the circumcenter, the incenter andone vertex.

Crux 292.S4.148*. (Trigg)

Fold a square piece of paper to form four creases that determine angles withtangents of 1, 2, and 3.

Crux 309.S4.200. (Peter Shor)

Let ABC be a triangle with a ≥ b ≥ c or a ≤ b ≤ c. Let D and E be themidpoints of AB and AC, and let the bisectors of angles BAE and BCDmeet at R. Prove that

(a) AR ⊥ CR if and only if 2b2 = c2 + a2;(b) if 2b2 = a2 + c2, then R lies on the median from B.Is the converse of b true? See Crux 210.S4.13.

Crux 313.S4.207*. (Leon Bankoff)

In an RMS triangle ABC (that is, a triangle in which 2b2 = c2 + a2), provethat GK, the join of the centroid and the symmedian point, is parallel tothe base b.

Crux 315.S4.227. (O.Ramos)

Prove that if two points are conjugate with respect to a circle, the sum oftheir powers is equal to the square of the distance between them.

Crux 317.S4.230*. (J.G.Propp)

In triangle ABC, let D and E be the trisection points of side BC with Dbetween B and E, let F be the midpoint of side AC, and let G be themidpoint of side AB. Let H be the intersection of segments EG and DF .Find the ratio EH : HG by means of mass points or otherwise.


Crux 318.S4.231. (C.A.Davis)

Given any triangle ABC, thinking of it as in the complex plane, two pontsL and N may be defined as the stationary values of a cubic that vanishesat the vertices A, B, C. Prove that L and N are the foci of the ellipse thattouches the sides of the triangle at their midpoints, which is the inscribedellipse of maximal area.

See also Crux 659.


The sides of triangle ABC are trisected by the points P1, P2, Q1, Q2, R1,R2 as shown in the figure below. Show that

(a) P1Q1R1 ≡ P2Q2R2;(b) |P1Q1R1| = 1

3 |ABC|,(c) the sides of triangles P1Q1R1 and P2Q2R2 trisect each other;(d) if M1 is the midpoint of AB, then C, S, T , M1 are collinear.

Crux 322.S4.254. (H.Sitomer)

In parallelogram ABCD, angle A is acute and AB = 5. Point E is on ADwith AE = 4 and BE = 3. A line through B, perpendicular to CD, inter-sects CD at F . If BF = 5, find EF . A geometric solution (no trigonometry)is desired.


Crux 325.S4.258*;5.49. (B.C.Rennie)

It is well known that if you put two pins in a drawing board and a loopof string around them you can draw an ellipse by pulling the string tightwith a pencil. Now suppose that instead of two pins you use an ellipse cutout from plywood. Will the pencil in the loop of string trace out anotherellipse?

Crux 330.S4.263*. (M.S.Klamkin)

It is known that if any one of the following three conditions holds for a giventegrahedron, then the four faces of the tetrahedron are mutually congruent,i.e., the tetrahedron is isosceles:

(1) The perimeter of the four faces are mutually equal.(2) The areas of the four faces are mutually equal.(3) the circumcircles of the four faces are mutually congruent.Does the condition that the incircles of the four faces be mutually con-

gruent, also, imply that the tetrahedron be isosceles?See also Crux 478.S6.217*.

Crux 338.S4.290*. (W.A.McWorter)

Can one locate the center of a circle with a VISA card?

Crux 353.S5.56*. (O.Ramos)

Prove that if a triangle is self polar with respect to a parabola, its nine -point circle passes through the focus.

Crux 363.S5.110*. (R.H.Eddy)

The following generalization of the Fermat point is known: if similar isoscelestriangles BCA′, CAB′, ABC ′ are constructed externally to triangle ABC,then AA′, BB′, CC ′ are concurrent.

Determine a situation in which AA′, BB′, CC ′ are concurrent if theconstructed triangles are isosceles but not similar.

Crux 364.S5.113*. (S.R.Mandan)

In the euclidean plane, if xi1, (x = a, b; i = 0, 1, 2), are the 2 triads of

perpendiculars to a line p from 2 triads of points X1i (X = A,B) on p and

(X) a pair of triangles with vertices Xi on x − 1i and sides xi opposite Xi


such that the three perpendiculars to bi from A1i concur at a point G, then

it is true for every member of the 3-parameter family f(B) of triangles like(B); andthe3perpendicularsfromB1

i to the sides ai of any memebr of the3-parameter family f(A) of triangles like A concur at a point G′ if and onlyif

A10A

11

A11A

12

=B1

0B11

B11B

12

.

Crux 365.S5.114*. (K.Satyanarayana)

A scalene triangle ABC is such that the external bisectors of angles B andC are of equal length. Given the lengths of sides b, c, (b > c), find the lengthof the third side a and show that its value is unique.

Crux 379.S5.149*. (P.Arends)

Construct a triangle ABC, given angle A and the lengths of side a and ta(the internal bisector of angle A).

Editor’s Remark: This can be found in Casey, Sequel, p.80. It alsoappeared in the Monthly, in 1906, 1931, and 1974. See AMM E2499.

Crux 383.S5.174*. (D.Skolowsky)

Let ma, mb, mc be the medians of triangle ABC. Prove that(a) if ma : mb : mc = a : b : c, then triangle ABC is equilateral;(b) if mb : mc = c : b, then either (i) b = c or (ii) quadrilateral AEGF is

cyclic;(c) if both (i) and (ii) hold in (b), then triangle ABC is equilateral.

Crux 386.S5.179. (Francine Bankoff)

A square PQRS is inscribed in a semicircle (O) with PQ falling along di-ameter AB. A right triangle ABC, equivlaent to the square, is inscribed inthe same semicircle with C lying on the arc RB. Show that the incenter Iof triangle ABC lies at the intersection of SB and RQ, and that

RI

IQ=SI

IB=

1 +√

52

, the golden ratio.


Crux 388.S5.201. (W.J.Blundon)

Prove that the line containing the circumcenter and the incenter of a triangleis parallel to a side of the triangle if and only if

s2 =(2R − r)2(R+ r)

R− r.

Crux 397.S5.234. (J.Garfunkel)

Given is triangle ABC with incenter I. Lines AI, BI, CI are drawn tomeet the incircle (I) for the first time in D, E, F respectively. Prove that√

3(AD+BE+CF ) is not less than the perimeter of the triangle of maximumperimeter that can be inscribed in circle (I).

Crux 412.S5.300* (K.Satyanarayana)

The sides BC, CA, AB of triangle ABC are produced respectively to D, E,F so that CD = AE = BF . Show that triangle ABC is equilateral if (andonly if) DEF is equilateral.

Crux 414.S5.304*. (B.C.Rennie)

A few years ago a distinguished mathematician wrote a book saying that thetheorems of Ceva and Menelaus were dual to each other. Another distin-guished mathematician reviewing the book wrote that they were not dual.Explain why they were both right, or if you are feeling in a sour mood, whythey were both wrong.

Crux 415.S5.306*. (A.Liu)

Is there a euclidean construction of a triangle given two sides and the radiusof the incircle?


Let A0BC be a triangle aand a a positive number les than 1. Construct P1

on A0B so hat A0P1/A0B = a. Construct A1 on P1C so that P1A1/P1C = a.Inductively construct Pn+1 on AnB so that AnPn+1/AnB = a and constructAn+1 on Pn+1C so that Pn+1An+1/Pn+1C = a. Show htat lla the Pi are ona line and all the Ai are on a line, the two lines being parallel.


Crux 419.S6.19. (G.Ramanaiah)

A variable point P describeds the ellipse

x2

a2+y2

b2= 1.

Does it make sense to speak of “the mean distance of P from a focus S” ?If so, what is this mean distance?

Crux 420.S6.21*. (J.A.Spencer)

Given an angle AOB, find an economical euclidean construction that willquadrisect the angle.

Proposer: 5 euclidean operations suffice.

Crux 422.S6.24*. (Pedoe)

The line and m are parallel edges of a strip of paper and P1, Q1 are pointson and m respectively. Fold P1Q1 along and crease, obtaining P1Q2 asthe crease. Fold P1Q2 along m and crease, obtaining P2Q2. Fold P2Q2 along and crease, obtaining P2Q3. If the process is continued indefinitely, showthat the triangle PnPn+1Qn+1 tends towards an equilateral triangle.

Crux 423.S6.26*. (J.Garfunkel)

ta ≤ cos2 A

2cos

B − C

2≤ ma.

Crux 427.

A corridor of width a intersects a corridor of width b to form an L. Arectangular plate is to be taken along one corridor, around the corner andalong the other corridor with the plate being kept in a horizontal plane.Among all the plates for which this is possible, find those of maximum area.

Crux 428.S6.50. (J.A.Spencer)

Let AOB be a right - angled triangle with legs OA = 2OB. Use it to find aneconomical euclidean construction of a regular pentagon whose side is notequal to any side of AOB. “Economical” means here using the smallestpossible number of euclidean operations: setting a compass, striking an arc,drawing a line.


Crux 435.S6.60. (J.A.H.Hunter)

In rectangle ABDF , AC = 125, CD = 112, DE = 52, as shown in thefigure,and AB, AD, AF are also integral. Evaluate EF .


A circle is inscribed in a square ABCD. A second circle on diameter BEtouches the first circle. Show that AB = 4BE.

Crux 445.S6.92. (Jordi Dou)

Consider a family of parabolas escribed to a given triangle. To each parabolacorresponds a focus F and a point S of intersection of the lines joining thevertices of the triangle to the points of contact with the opposite sides. Provethat all lines FS are concurrent.


Crux 450.S6.120*;214. (Andy Liu)

Triangle ABC has a fixed base BC and a fixed inradius. Describe thelocus of A as the incircle rolls along BC. When is AB of minimal length(geometric characterization desired)?

Crux 454.S6.125*. (R.R.Tiwari)

(a) Is there a euclidean construction for a triangle ABC given the lengthsof its internal angle bisectors ta, tb, tc ?

(b) Find formulas for the sides a, b, c in terms of ta, tb, tc. See also Crux749.

See also Crux 749.

Crux 456.S6.128. (O.Ramos)

Let ABC be a triangle and P any point in the plane. Triangle MNO isdetermined by the feet of the perpendicular from P to the sides, and triangleQRS is determined by the cevians through P and the circumcircle of triangleABC. Prove that triangle MNO and QRS are similar.


Crux 462.S6.162. (H.Charles)

Soient A, B, C les angles d’un triangle. Montrer que

det

⎛⎝ tan A

2 cosA 1tan B

2 cosB 1tan C

2 cosC 1

⎞⎠ = 0.


Construct an equilateral triangle so that one vertex is at a given point, asecond vertex is on a given line, and the third vertex is on a given circle.

See also Crux 545.

Crux 464.S6.185.*. (J.C.Fisher and E.L.Koh)

(a) If the two squares ABCD and AB′C ′D′ have vertex A in common andare taken with the same orientation, then the centers of the squares togetherwith the midpoints of BD′ and B′D arethe vertices of a square.

(b) What is the analogous theorem for regular n−gons?

Crux 466.S6.188*. (R.Fischler)

Soient AB et BC deux arcs d’un cercle tels que arc AB >arc BC, et soitD le point de milieu de l’arc (voir la figure). Si DE ⊥ AB, montrer queAE = EB +BC. [Ce theoreme est atribue a Archimede].

Crux 472.S6.196. (J.Dou)

Construire un triangle connaissant le cote b, le rayon R du cercle circonscrit,et tel que la droite qui joint les centres des cercles inscrit et circonscrit soit


parallele au cote a.


Construct an isosceles right triangle such that the three vertices lie eachon one of three concurrent lines, the vertex of the right angle being on theinside line.

Crux 478.S6.219*;11(6)189. (M.S.Klamkin)

If the circumcircles of the four faces of a tetrahedron are mutually congruent,then the circumcenter O of the tetrahedron and its incenter I coincide.

An editor’s comment following Crux 330 claims that the proof this the-orem is easy. Prove it.

Crux 483.S6.226*. (S.Collings)

Let ABCD be a convex quadrilateral, AB DC intersecting at F and AD,BC intersecting at G. Let IA, IB , IC , ID bethe incenters of triangles BCD,CDA, DAB, and ABC respectively.

(a) ABCD is a cyclic quadrilateral if and only if the internal bisectorsof the angles at F and G are perpendicular.

(b) If ABCD is cyclic, then IAIBICID is a rectangle. Is the conversetrue?

Crux 485.S6.256. (M.S.Klamkin)

Given three concurrent cevians of a triangle ABC intesecting at a pointP , we construct three new points A′, B′, C ′ such that AA′ = k · AP ,BB′ = k ·BP , CC ′ = k ·CP , where k > 0 and k = 1, and the segments aredirected. Show that A, B, C, A′, B′,C ′ lie on a conic if and only if k = 2.

See also Crux 672.


Given a point P within a given angle, construct a line through P such thatthe segment intercepted by the sides of the angle has minimum length.


Crux 492.S6.291*;7.50,277;8.79. (D.Pedoe)

(a) A segment AB and a rusty compass of span r ≥ 12AB are given. Show

how to find the vertex C of an equilateral triangleABC, using, as few timesas possible, the rusty compass only.

(b) Is the construction possible when r < 12AB?

See also Crux 592.

Crux 493.S6.294*;7.50. (R.C.Lyness)

(a) A, B, C are the angles of a triangle. Prove that there are positive x, y,z, each less than 1

2 , such that

y2 cotB

2+ 2yz + z2 cot

C

2= sinA,

z2 cotC

2+ 2zx+ x2 cot

A

2= sinB,

y2 cotA

2+ 2xy + y2 cot

B

2= sinC.

(a) In fact, 12 may be replaced by a smaller k > 0.4. What is the least

value of k?Note relation to the Malfatti circles.

Crux 504.S7.25*. (L.Bankoff)

Given is a triangle ABC and its circumcircle. Find a euclidean constructionfor a point J inside the triangle such that, when the chords AD, BE, CFare all drawn from J , then DEF is equilateral.


It is known from an earlier problem in this journal [Crux 14.S1.28] that ifa, b, c are the sides of a triangle, then so are 1

b+c ,1

c+a , 1a+b . Show more

generally that if a1, a2, . . . , an are the sides of a polygon then for k = 1, 2,,. . . , n,

n+ 1S − ak

≥n∑

i=1,i=k

1S − ai

≥ (n− 1)2

(2n− 3)(S − ak),

where S = a1 + a2 + · · · + an.


Crux 515.S7.57*. (Ngo Tan)

Given is a circle with center O and an inscribed triangle ABC. DiametersAA′, BB′, CC ′ are drawn. The tangent at A′ meets BC in A′′, the tangentat B′ meets CA om B′′, and the tangent at C ′ meets AB in C ′′. Show thatthe points A′′, B′′, C ′′ are collinear.


hb

mc+hc

ma+ha

mb≤ 3,



If two chords of a conic are mutually bisecting, prove that the conic cannotbe a parabola.

Crux 529.S7.91*. (J.T.Groenman)

The sides of a triangle ABC satisfy a ≤ b ≤ c.

sgn(2r + 2R− a− b) = sgn(2rc − 2R− a− b) = sgn(C − 90).



Let Ta, Tb, Tc denote the angle bisectors extended to the circumcircle oftriangle ABC. Prove that

TaTbTc ≥ 89

√3abc,

with equality attained in the equilateral triangle.

Crux 536.S7.122. (B.Leeds)

Through each of the midpoints of the sides of a triangle ABC, lines aredrawn making an acute angle θ with the sides. These lines intersect to forma triangle A′B′C ′. Prove that A′B′C ′ is similar to ABC and find the ratioof similarity.

Crux 540.S7.127*;240. (Leon Bankoff)

Professor Euclide Paracelso Bombasto Umbugio has once again retired to histour d’ivoire where he is now delving into the supersophisticated intricaciesof the works of Grassmann, as elucidated by Forder’s Calculus of Extension.His goal is to prove Neuberg’s Theorem:

If D, E, F are the centers of squares described externally on the sides ofa triangle ABC, then the midpoints of these sides are the centers of squaresdescribed internally on thesides of triangle DEF .


Help the dedicated professor emerge from his self - imposed confinementand enjoy the thrill of hyperventilation by showing how to solve his problemusing only high school, synethetic, euclidean, “plain” geometry.

Crux 544.S7.150*. (N.N.Murty)

2∑

sinB

2sin

C

2≤∑

sinA

2with equality if and only if the triangle is equilateral.

See Klamkin’s solution, relating to AMM S23.


Given three concentric circles, construct an equilateral triangle having onevertex on each circle.


If three equal cevians of a triangle divide the sides in the same ratio andsame sense, must the triangle be equilateral?

Crux 554.S7.184*;10.197. (G.C.Giri)

A sequence of triangle is defined as follows. 0 iis a given triangle, andfor each triangle n in the sequence, the vertices of n+1 are the pointsof contact of the incircle of n with its sides. Prove that n tends to anequilateral triangle as n→ ∞.

See also Crux 463.

Crux 560.S7.243*. (B.C.Rennie)

Take a complete quadrilateral. On each of the three diagonals as diameterdraw a circle. Prove that these three circles are coaxal.


Given is a circle γ with center O and diameter of length d, two distinctpoints P and Q not collinear with O, and a segment of length , where0 ≤ ≤ d. Construct a circle through P and Q which meets γ in pointsCand D such that chord CD has length ell.



In an acute - angled triangle ABC, the altitude issued from vertex A [B, C]meets the internal bisector of angle B [C, A] at P [Q, R]. Prove that

AP · BQ · CR = AI ·BI · CI,

where I is the incenter of triangle ABC.

Crux 567.S7.214. (G.Tsintsifas)

A moving equilateral triangle has its vertices A′, B′, C ′ on the sides BC,CA, AB respectively of a fixed triangle ABC. The regular tetrahedronMA′B′C ′ has its vertex M always on the same side of the plane ABC. Findthe locus of M .

Crux 569.S7.216. (C.W.Trigg)

Using euclidean geometry, show that the planes perpendicualr to a spacediagonal of a cube at its trisection points contain the vertices of the cubenot on that diagonal.

Crux 574.S7.247. (J.Dou)

Given five points A, B, C, D, E, construct a straight line such that thethree pairs of straight lines (AD,AE), (BD,BE), (CD,CE) intercept equalsegments on .


In the figure, the diameter PQ ⊥ BC and chord AT ⊥ BC. Show that

AQ

PQ=AB +AC

PB + PC=TB + TC

QB +QC.


Crux 584.S7.290;8.16,51*,107;9.23. (F.G.B.Maskell)

If a triangle is isosceles, then its centroid, circumcenter, and the center ofan escribed circle are collinear. Prove the converse.


Given is a triangle ABC with internal angle bisectors ta, tb, tc, and mediansma, mb, mc to sides a, b, c respectively. If

ma ∩ tb = P, mb ∩ tc = Q, mc ∩ ta = R,

and L, M , N are the midpoints of the sides a, b, c, prove that,

AP

PL· BQQM

· CRRN

= 8.

See also Crux 685 and 790.

Crux 589.S7.307. (Ngo Tan)

In a triangle ABC with seimperimeter s, sides of lengths a, b, c, and mediansof lengths ma, mb, mc, prove that

(a) There exists a triangle with sides lengths a(s− a), b(s− b), c(s− c).(b) m2

aa2 + m2

bb2 + m2

cc2 ≥ 9

4 , with equality if and only if the triangle isequilateral.

Crux 592.S7.310. (L.F.Meyers)

(a) Given a segment AB of length , and a rusty compass of fixed openingr, show how to find a point C such that the length of AC is the meanproportiaonal between r and , by use of the rusty compass only, if 1

4 ≤r ≤ , but r = 1

2.(b) Show that the construction is impossible if r = 1

2.(c) Is the construction possible if r < 1

4 or r > ?


Given a triangle ABC and a segment PQ on side BC, find, by euclideanconstruction, segments RS on side CA and TU on side AB such that, ifequilateral triangles PQJ , RSK, and TUL are drawn outside the giventriangle, then JKL is an equilateral triangle.



ABC is a triangle with sides of lengths a, b, c and semiperimeter s. Provethat

cos4 12A+ cos4 1

2B + cos4 1

2C ≤ s3

2abcwith equality if and only if the triangle is equilateral.


A1B1C1D1 is a convex quadrilateral inscribed in a circle and M1, N1, P1,Q1 are the midpoints of the sides B1C1, C1D1, D1A1, A1B1 respectively.The chords A1M1, B1N1, C1P1, D1Q1 meet the circle again in A1, B2,C2, D2 respectively. Quadrilaterals A3B3C3D3 is formed from A2B2C2D2

as the latter was fromed from A1B1C1D1, and the procedure is repeatedindefinitely. Prove that quadrilateral AnBnCnDn tends to a square as n→∞. What happens if A1B1C1D1 is not convex?

Crux 613.S8.55*,138*. (J.Garfunkel)

If A+B + C = 180circ, prove that

cos12(B − C) + cos

12(C −A) + cos

12(A−B) ≥ 2√

3(sinA+ sinB + sinC).

Here, A, B, C are not necessarily the angles of a triangle, but you mayassume that they are if it is helpful to achieve a proof with calculus.

Crux 614.S56. (J.T.Groenman)

Given is a triangle with sides of length a, b, c. A point P moves inside thetriangle in such a way that the sum of the squares of its distances to thethree vertices is a constant (= k2). Find the locus of P .

This is the circle, center G, radius 13

√3k2 − (a2 + b2 + c2).

Crux 615.S8.57. (G.P.Henderson)

Let P be a convex n−gon with vertices E1, . . . , En, perimeter L and area A.Let 2θi be the measure of the interior angle at vertex Ei and set C =

∑cot θi.

Prove thatL2 − 4AC ≥ 0

and characterize the convex n−gons for which equality holds.


Crux 618.S8.82*,175. (J.A.H.Hunter)

For i = 1, 2, 3, let Ii be the centers and ri the radii of the three Malfatticircles of a triangle ABC. Calculate the side lengths of the triangle.

Comment by Dimitris Vathis, Greece: the inverse Malfatti problemappears and is solved in Pallas, Great Algebra, Athens, 1957 (in Greek),pp.103–104.


If PQR is the equilateral triangle of smallest area inscribed in a given tri-angle ABC, with P on BC, Q on CA, and R on AB, prove or disprove thatAP , BQ, CR are concurrent.

Crux 624.S8.109*. (Dmitry P. Mavlo)

ABC is a given triangle of area K, and PQR is the equilateral triangle ofsmallest area K0 inscribed in triangle ABC, with P on BC, Q on CA, andR on AB.

(a) Find the ratio λ = KK0

≡ f(A,B,C) as a function of the angles ofthe given triangle.

(b) Prove that λ attains its minimum value when the given triangle ABCis equilateral.

(c) Give a euclidean construction of triangle PQR for an arbitrary giventriangle ABC.

Crux 628.S8.115*. (R.H.Eddy)

Given a triangle ABC with sides a, b, c, let Ta, Tb, Tc denote the anglebisectors extended to the circumcircle of the triangle. If R and r are thecirucm- and in-radii of the triangle, prove that

Ta + Tb + Tc ≤ 5R+ 2r,

with equality just when the triangle is equilateral.

Crux 633.S8.120.*. (J.Aldins, J.S.Kline, and Stan Wagon)

It follows from the Wallace - Bolyai - Gerwien Theorem of the early 19thcentury that any triangle may be cut up into pieces which may be rearrangedusing only translations and rotations to form the mirror of the given triangle.


This problem once appeared in a Moscow Mathematical Olympiad. Showthat such a dissection may be effected with only two straight cuts.


Crux 635.S8.139. (Dan Sokolowsky)

In the figure below, O is the circumcenter of triangle ABC, and PQR ⊥ OA,PST ⊥ OB. Prove that

PQ = QR⇐⇒ PS = ST.

Crux 637.S8.143*. (J.Bhattacharya)

Given a, b, c > 0, and 0 < A, B, C < π, and

a = b cosC + c cosB, b = c cosA+ a cosC, c = a cosB + b cosA,

prove thata

sinA=

b

sinB=

c

sinCand A+B + C = π.


crux 643.S8.153. (J.T.Groenman)

For i = 1, 2, 3, a given triangle has vertices Ai, interior angles αi, and sidesai. Segments AiDi, which terminates in ai, bisects angles αi; mi is theperpendicular bisector of AiDi; and Ei the intersection of ai and mi. Provethat

(a) the three points Ei are collinear;(b) the three segments EiAi are tangent to the circumcircle of the tri-

angle;(c) if pi is the length of AiEi, and if a1 ≤ a2 ≤ a3, then

1p3

+1p1

=1p2.

The points Ei are the centers of the Apollonian circles. The line contain-ing these centers is called the Lemoine axis of the triangle [Court, p.253].


If I is the incenter of triangle ABC, and the lines AI, BI, CI meet thecircumcircle again at D, E, F , prove that

AI

ID+BI

IE+CI

IF≥ 3.


The ratio AI : ID = b+ c− a : a. The sum is∑ a

b + ba − 3.

See also AMM S23 (1981) 536–537.

See also MG 1119, where I is replaced by G, with equality, and MG1120, where I is replaced by an arbitrary point P , and asks for the set ofpoints for which the inequality holds.

Crux 646.S8.175. (J.C.Fisher)

Let M be the midoint of a segment AB.(a) What is the locus of a point the product of whose distances from A

and B is the square of its distance from M?(b) In a circle γ through M and B, the three chords BP , BP ′, and BM

satisfyBP · BP ′ =

√2BM.

Prove that the tangent to γ at M meets the lines BP and BP ′ (extended)in points X and Y , respectively, that are equidistant from M . Note thatthis fact suggests a construction for the locus of part (a), since X and Ysatisfy

XA ·XB = Y A · Y B = XM2.

(c) What is the locus of a point for which the absolute value of thedifference of its distances from A and B equals

√2 times its distance from

M?

Crux 648*.S8.180*. (J.Garfunkel)

Given are a triangle ABC, its centroid G, and the pedal triangle PQR ofits incenter I. The segments AI, BI, CI meet the incircle in U , V , W ; andthe segments AG, BG, CG meet the incircle in D, E, F . Let ∂ denote theperimeter of a triangle and consider the segment

∂PQR ≤ ∂UVW ≤ ∂DEF.

(a) Prove the first inequality.(b) Prove the second inequality.


AI : DI = csc A2 : 1; AD : DI = csc A

2 − 1 : 1. It follows that

D = sinA

2[A+ (csc

A

2− 1)I].

Similarly,

E = sinB

2[B + (csc

B

2− 1)I],

andF = sin

C

2[C + (csc

C

2− 1)I].

The area of triangle DEF is

12s

sinA

2sin

B

2sin

C

2· ABC[2s+ a(csc

C

2− 1) + b(csc

B

2− 1) + c(csc

A

2− 1)]

=12s

sinA

2sin

B

2sin

C

2· ABC[c csc

C

2+ b csc

B

2+ a csc

A

2]

=12s

[a sinB

2sin

C

2+ b sin

C

2sin

A

2+ c sin

A

2sin

B

2].

=

Crux 652.S8.188*. (W.J.Blundon)

Let R, r, s represent respectively the circumradius, the inradius, and thesemiperimeter of a triangle with angles α, β, γ. It is well known that

∑sinα =

s

R,

∑cosα =

R+ r

R,

∑tanα =

2rss2 − 4R2 − 4Rr − r2

.

As for half angles, it is easy to prove that

∑tan

α

2=

4R+ r

s.

Find similar expressions for∑

cos α2 and

∑sin α

2 .See a paper of Rabinowitz in a later issue (around 1988).

Crux 653.S8.190*. (G.Tsintsifas)

For every triangle ABC, show that

∑cos2 B − c

2≥ 24

∏sin

A

2,




P is an interior point of a covex region R bounded by the arcs of twointersecting circles C1 and C2. Construct through P a “chord” UV of R,with U on C1 and V on C2, such that PU · PV is a minimum.


A quadrilateral ABCD is circumscribed about a circle γ, and

AH = AE = a, BE = BF = b, CF = CG = c, DG = DH = d.

Prove thatIA

IC=a

b,

IB

ID=b

d.

Crux 659.S8.215. (Leon Bankoff)

If the line joining the incenter I and the circumcenter O of triangle ABC isparallel to the side BC, it is known from Crux 318 that

s2 =(2R − r)2(R+ r)

R− r.

In the same configuration, cosB + cosC = 1, and the internal bisector ofangle A is perpendicular to the line joining I to the orthocenter H (MG758). Prove the following additional properties:

(a) If the internal bisector of angle A meets the circumcircle in P , showthat AI : IP = cosA.


(b) The circumcircle of triangle AIH is equal to the incircle of triangleABC.

(c) AI · IP = 2Rr =√R ·AI ·BI · CI.

(d) sin2 B2 + sin2 C

2 = 12 ; cos2 B

2 + cos2 C2 = 3

2 .(e) tan2 A

2 = R−rR+r .

Crux 660(corrected 7.274)S8.216. (Leon Bankoff)

Show that, in a triangle ABC with semiperimeter s, the line joining thecircumcenter and the incenter is parallel to BC if and only if

DL+ EM + FN = s tanA

2,

where L, M , N bisect the arcs BC, CA, AB of the circumcircle, and D, E,F bisects the sides BC, CA, AB respectively, of the triangle.

Crux 662.S8.218*. (Kaidy Tan)

An isosceles triangle has vertex A and base BC. Through a point F on AB,a perpendicular to AB is drawn to meet AC in E and BC produced in D.With square brackets denoting area, prove synthetically that

[AFE] = 2[CDE] ⇐⇒ AF = CD.


An isosceles trapezoid ABCD, with parallel bases AB and DC, is inscribedin a circle of diameter AB. Prove that AC > 1

2(AB +DC).

Crux 665.S8.221*,280. (J.Garfunkel)

If A, B,C, D are the interior angles of a convex quadrilateral ABCD, provethat √

2∑

cosA+B

4≤∑

cotA

2,

where the four term sum on each side is cyclic over A, B, C, D, with equalityif and only if ABCD is a rectangle.


Crux 666.S8.222. (J.T.Groenman)

The symmedians issued from vertices A, B, C of triangle ABC meet theopposite sides in D, E, F respectively. Through D, E, F , lines d, e, f aredrawn perpendicular to BC, CA, AB respectively. Prove that d, e, f areconcurrent if and only if ABC is isosceles.

[For what point P is the cevian triangle also a pedal triangle?]

More generally, if P = f : g : h, the perpendicular from the trace 0 : g;hto the side BC is the line

[(a2 + b2 − c2)h− (c2 + a2 − b2)g]x + 2abhy − 2acgz = 0.

Similarly, the equations of the other two perpendiculars can be written down.These three perpendiculars are concurrent if and only if

(b2+c2−a2)f(g2−h2)+(c2 +a2−b2)g(h2−f2)+(a2+b2−c2)h(f2−g2) = 0.

For the symmedian point K, this expression is

(a− b)(b− c)(c − a)(a+ b)(b+ c)(c+ a)(a2 + b2 + c2).

For the incenter, this is

(a− b)(b− c)(c − a)(a+ b+ c)2.

Crux 667.S8.248*. (Dan Sokolowsky)

A plane is determined by a line D and a point F not on D. Let C denotethe conic consisting of all those points P in the plane for which PF

PO = r,where PO is the distance from P to D and r is a given real number.

Given a line in the plane, show how to determine by elementary meansthe intersection (if any) of and C.


Crux 670.S8.251. (O.Bottema)

The points Ai, i = 1, 2, . . . , 6, no three of which are collinear, are the verticesof a hexagon. X0 is an arbitrary point other than A2 on line A1A2. Theline through X0 parallel to A2A3 intersects A3A4 in X1; the line through X1

parallel to A3A6 intersects A6A1 in X2; the line through X2 parallel to A5A6

intersects A4A5 in X3; and the line through X3 parallel to A2A5 intersectsA1A2 in X4.

(a) Prove the following closure theorem: if X0X1X2X3X4 is closed, i.e.,if X4 coincides with X0 for some point X0, then it is closed for any pointX0.

(b) Show that closure takes place if and only if the six points Ai lie ona conic.


Given four points P , A, B, C in a plane, determine points A′, B′, C ′ onPA, PB, PC respectively, such that

AA′

PA′ = tα,BB′

PB′ = tβ,CC ′

PC ′ = tγ,

where α, β, γ are given constants, and such that the hexagon ABCA′B′C ′

is inscribed in a conic.This generalizes Crux 485.

Crux 674(corrected 7.236).S8.256. (G.Tsintsifas)

Let ABC be a given triangle and A′B′C ′ its medial triangle. The incircleof the medial triangle touches its sides in R, S, T . IF the points P and Qdivide the perimeter of the original triangle into two equal parts, prove thatthe midpoint of segment PQ lies on the perimeter of triangle RST .

Crux 675.S8.257. (H.D.Ruderman)

ABCD is a skew quadrilateral and P , Q, R, S are points on sides AB, BC,CD, DA respectively. Prove that PR intersects QS if and only if

AP · BQ · CR ·DS = PB ·QC · RD · SA.


Crux 682.S8.287*;9.23*. (R.C.Lyness)

Triangle ABC is acute angled and 1 is its orthic triangle, 2 is the trian-gular hull of the three excircles aof ABC. Prove that the area of 2 is atleast 100 times that of 1.

Crux 683.S8.289*. (Kaidy Tan)

Triangle ABC has AB > AC, and the internal bisector of angle A meetsBC at T . Let P be any point other than T on line AT , and suppose linesBP , CP intersect lines AC, AB in D, E respectively. Prove that BD > CEor BD < CE according as P lies on the same side or on the opposite sideof BC as A.

Crux 685.S8.292.* (J.T.Groenman)

Given is a triangle ABC with internal angle bisectors and medians.

Crux 588 asks for a proof of

AP

PD· BQQE

· CRRF

= 8.

Establish here the inequality

AR

RX· BPPY

· CQQZ

≥ 8,



Crux 686.S8.294*;9.25. (C.W.Trigg)

Find the area of the region which is common to four quadrants that havethe vertices of a square as centers and a side of the square as a commonradius.

Answer: 1 −√3 + π

3 .


Let ma, mb, mc be the lengths of the medians to sides a, b, c of triangleABC, and let Ma, Mb, Mc denote the lengths of these medians extended tothe circumcircle of the triangle. Prove that

Ma

ma+Mb

mb+Mc

mc≥ 4.


Three concurrent circles with radical center R lie inside a given trianglewith incenter I and circumcircle O. Each circle touches a pair of sides ofthe triangle. Prove that O, R, and I are collinear.

See also Crux 2137.


Crux 695(corrected 8.30).S8.314. (J.T.Groenman)

For i = 1, 2, 3, Ai are the vertices of a triangle with sides ai, and excircleswith centers Ii touching ai at Bi. For j, k = i, Mi are the midpoints ofBjBk; and mi are the lines through Mi perpendicular to ai. Prove that themi are concurrent.

The midpoint of B2B3 is the point

M1 = (b+ c)(s − a) : b(s− b) : c(s − c).

The perpendicular to the line BC is

−a(b− c)(a3 − a2b− ab2 + b3 − a2c− 2abc− b2c− ac2 − bc2 + c3)x−b(a3b− a2b2 − ab3 + b4 − a3c− 4a2bc− ab2c+ 2b3c+ a2c2 + abc2 + ac3 − 2bc3 − c4)y+c(−a3b+ a2b2 + ab3 − b4 + a3c− 4a2bc+ ab2c− 2b3c− a2c2 − abc2 − ac3 + 2bc3 + c4)z = 0.

The other two perpendiculars can be written down. These three perpen-diculars intersect at the point


[a(a5b− a4b2 − 2a3b3 + 2a2b4 + ab5 − b6 + a5c+ 2a2b3c− ab4c− 2b5c− a4c2

+b4c2 − 2a3c3 + 2a2bc3 + 4b3c3 + 2a2c4 − abc4 + b2c4 + ac5 − 2bc5 − c6)].


(a) 34 + 1

4

∑cos 1

2(B − c) ≥∑cosA;

(b)∑a cos 1

2(B − C) ≥ s(1 − 2rR ).

Crux 699.S8.320. (C.W.Trigg)

A quadrilateral is inscribed in a circle. One side is a diameter of the circleand the other sides have lengths 3, 4, 5. What is the length of the diameterof the circle?

Crux 700.S9.25.* (Jordi Dou)

Construct the center of the ellipse of minimum eccentricity circumscribedto a given convex quadrilateral.

Crux 702.S8.323*;9.144*. (Tsintsifas)

Given three distinct points A1, B1, C1 on a circle, and three arbitrary realnumbers , m, n adding to 1, show how to determine a point M such thatif A1M , B1M , C1M meet the circle again at A,B,C, then

MBC = ABC, MCA = mABC, MAB = nABC.


Crux 703.S9.28. (S.Rabinowitz)

A right triangle has legs AB = 3 and AC = 4. A circle γ with center Gis drawn tangent to the two legs and tangent internally to the circumcircleof the triangle, touching that circumcircle in H. Find the radius of γ andprove that GH is parallel to AB.

Crux 708.S9.49 (Murty, Blundon)

(a) Prove that ∑(2a− s)(b− c)2 ≥ 0,

with equality just when the triangle is equilateral.(b) Prove that the inequality in (a) is equivalent to each of the following

4s∑

a2 ≥ 3(∑

a3 + 3abc),s2 ≥ 16Rr − 5r2.

Blundon first showed that these inequalities are equivalent, and estab-lished a generalization of (1) using Schur’s inequality: if t is any real numberand x, y, z > 0, then

xt(x− y)(x− z) + yt(y − z)(y − x) + zt(z − x)(z − y) ≥ 0,


Crux 709.S9.50.

ABC is a triangle with incenter I, and DEF is the pedal triangle of thepoint I with respect to the sides of ABC. Show that it is always possible


to construct four circles each of which is tangent to each of the circumcirclesof triangles ABC, EIF , FID, DIE, provided that ABC is not equilateral.


Crux 712.S9.56*. (D.Aitken)

Prove that AB = CD in the figure below.

Crux 715.S9.58. (Murty)

Let k be a real number, n an integer. (a) Prove that

8k(sin nA+ sinnB + sinnC) ≤ 12k2 + 9.

(b) Determine for which k equality is possible in (a), and deduce that

| sinnA+ sinnB + sinnC| ≤ 3√

32.

Crux 717.S9.64*. (J.T.Groenman and D.J. Smeenk)

Let P be any point in the plane of (but not on a side of) triangle ABC.If Ha, Hb, Hc are the orthocenters of triangles PBC, PCA, PAB respec-tively, prove that [ABC] = [HaHbHc], where the brackets denote the areaof triangle.


Editor’s note: This appears in Casey’s Trigonometry, p.146, where theproblem is credited to J. Neuberg.

Lemma

If ABCDEF is a hexagon (not necessarily convex) such that AB//DE,BC//EF , and CD//FA, then [ACE] = [BDF ].

Crux 718.S9.82. (Tsintsifas)

ABC is an acute triangle with circumcenter O. The lines AO,BO,COintersect BC,CA,AB at A1, B1, C1 respectively. Show that

OA1 +OB1 +OC1 ≥ 92.


On the sides AB and AC of a triangle ABC as bases, similar isoscelestriangles ABE and ACD are drawn outwardly. If BD = CE, prove ordisprove that AB = AC.

Generalization by O.Bottema: If similar triangles ABE and ACD aredrawn outwardly, BD = CE if and only if AB = AC or if triangles ABEand ACD are isosceles with vertex at A.

Crux 723.S9.91*,147. (Tsintsifas)

Let AG, BG, CG meet the circumcircle again in A′, B′, C ′ respectively.Prove that


(a) GA′ +GB′ +GC ′ ≥ AG+BG+ CG,(b)

∑ AGGA′ = 3,

(c) GA′ ·GB′ ·BC ′ ≥ AG ·BG · CG.

Crux 724.S9.92*. (H.Ahlburg)

Let C +A = 2B. Show that(a) sin(A−B) = sinA− sinC.(b) a2 − b2 = c(a − c). (c) A,C,O, I,H, Ib all lie on a circle, radius R.

Furthermore, if this circle meets the lines AB and BC again at A′ and C ′,then AA′ = CC ′ = |c− a|.

(d) OI = OH (Crux 739 asks for the converse).(e) s =

√3(R+ r) (Crux 260).

(f) OIb = HIb.(g) The nine-point center N lies on the internal bisector of B.

Crux 727.S9.115*,180*. (J.T.Groenman)

Let ta and tb be the symmedians issued from vertices B and C of ABCand terminating in the opposite sides b and c respectively. Prove that tb = tcif and only if b = c.

See PME 213.S70S.88.

Crux 728.S9.116*. (S.Rabinowitz)

Let E(P,Q,R) denote the ellipse with foci P , Q, which passes throughR. If A, B, C are distinct points in the plane, prove that no two ellipsesE(B,C,A), E(C,A,B), and E(A,B,C) can be tangent.

See also Crux 1063.

Crux 732.S9.119. (Groenman)

Given a fixed triangle and a varying circumscribing triangle determined byangle φ.

(a) Find a formula for the ratio of similarity λ = λ(φ) = B′C′BC .

(b) Find the maximal value λm of λ as φ varies in [0, π), and show howto construct the triangle when λ = λm.

(c) Prove that λm ≥ 2, with equality must when the given triangle isequilateral.


Crux 733.S9.121,149,210. (Garfunkel)

Let rm be the inradius of the triangle with sides the medians of a giventriangle. Prove or disprove

rm ≤ 3abc4(a2 + b2 + c2)

,

with equality just when the original triangle is equilateral.See also Crux 835.

Crux 735.S9.123*. (S.C.Chan)

In a given circle inscribe a triangle so that two sides may pass through twogiven points and the third side be parallel to a given straight line.

Crux 739.S9.153*,210. (G.C.Giri)

Prove that if I is equidistant from O and H, then one of the angles is 60.


A point M lies on the disc ω with diameter OG,. The lines AM , BM , CMmeet the circumcircle again in A′, B′, C ′ respectively, and G′ is the centroidof A′B′C ′. Prove that

(a) M does not lie in the interior of disc ω′ with diameter OG′;(b) ABC ≤ A′B′C ′.

Crux 745.S9.187*. (R.Izard)

In the adjoining figure, ABC and DEF are both equilateral, and anglesBAD, CBE, ACF are all equal. Prove that the triangles ABC and DEFhave the same center.

See also MG 1161.


Given are two concentric circles and a triangle ABC inscribed in the outercircle. A tangent to the outer circle at A is rotated about A in the counter-clockwise sense until it first touches the inner circle, say at P . The procedureis repeated at B and C, resulting in points A and R respectively, on the innercircle. Prove that PQR is directly similar to ABC.



Let ABC be a triangle inscribed in a circle with center O. The segmentsBC, CA, AB are divided internally in the same ratio by points A1, B1, C1

so thatBA1 : A1C = CB1 : B1A = AC1 : C1B = λ : µ,

where λ+µ = 1. A line through A1 perpendicular to OA meets the circle intwo points, one of which, Pa, lies on the arc CAB, and other points Pb andPc are determined analogously by lines through B1 and C1 perpendicular toOBand OC. Prove that

AP 2a +BP 2

b +CP 2c = a2 + b2 + c2,

independently of λ and µ.Investigate the situation if the word “internally” is replaced by “exter-

nally”.

Crux 749.S9.190,280* ( R.R.Tiwari)

Solve the system of equations

yz(x+ y + z)(y + z − x)(y + z)2

= a2,

zx(x+ y + z)(z + x− y)(z + x)2

= b2,

xy(x+ y + z)(x+ y − z)(x+ y)2

= c2.


Crux 752.S9.212. (Groenman)

For i = 1, 2, 3, let Ai be the vertices of a triangle with angles αi, sides ai,circumcenter O and inscribed circle γ. The lines AiO intersect γ in Pi andQi.

(a) Prove that

P1Q1 : P2Q2 : P3Q3 = f(cosα1) : f(cosα2) : f(cosα3),

where f(x) is a function to be determined.(b) Prove or disprove that α2 = α3 if and only if P2Q2 = P3Q3.

Crux 755.S9.244*. (L.Csirmaz)

Find the locus of points with coordinates

(cosA+ cosB + cosC, sinA+ sinB + sinC)

(a) if A,B,C are real numbers with A+B + C = π,(b) if A,B,C are angles of a triangle.

Crux 756.S9.217. (Lu YANG and Jingzhong ZHANG)

Given three vertices of a parallelogram, find the fourth vertex using only arusty compass.

Crux 757.S9.218. (A.Aeppli)

Given only two distinct points A and B, prove or disprove that the midpointof the segment AB can be found using only a rusty compass.


Find necessary and sufficient condition for the roots of a cubic equation tobe the vertices of an equilateral triangle.


Given are four congruent circle intersecting in a point O, and a quadrilat-eral ABCD circumscribing these circles with each side of the quadrilateraltangent to two circles. Prove that the quadrilateral ABCD is cyclic.



Given ABC, construct on AB and AC directly similar isosceles trianglesABX and ACY such that BY = CX. Prove that there are exactly twosuch pairs of isosceles triangles.

Crux 762(corrected 8.278);S9.249. (J.T.Groenman)

The internal bisectors of a triangle meet the opposite sides in D, E, Frespectively. The area of DEF is ′.

(a) Prove that3abc

4(a3 + b3 + c3)≤ ′

≤ 14.

(b) If a = 5 and 3abc4(a3+b3+c3) = 5

24 , determine b and c, given that they areintegers.

Crux 766.S9.254*. (S.Rabinowitz)

Let ABC be an equilateral trinagle with center O. Prove that if P is avariable point on a fixed caircle with center O, then the triangle whose sideshave length PA,PB,PC has a constant area.


49

∑sinB sinC ≤

∏cos

B − C

2≤ 2

3

∑cosA.

Crux 770.S9.285. (K.Satyanarayana)

Let P be an interior point of triangle ABC. Prove that

PA ·BC + PB · CA > PC ·AB.

Crux 777.S10.20*. (Bottema)

Let Q = ABCD be a convex quadrilateral with sides AB = a, BC = b,CD = c, DA = d, and area [Q]. The following theorem is well known: If Qhas both a circumcircle and an incircle, then [Q] =

√abcd.

Prove or disprove the following converse: If Q has a circumcircle and[Q] =

√abcd, then there exists a circle tangent to the four lines AB, BC,

CD, and DA.



Let ABC be a triangle with incenter I, the lines AI, BI, CI meeting itscircumcircle again in D, E, F respectively. If S is the sum and P the productof the numbers

ID

AI,

IE

BI,

IF

CI,

prove that 4P − S = 1.

Crux 782.S10.25*. (H.S.M.Coxeter)

(a) Sketch the plane cubic curve given by the parametric equations

x = α(β − γ)2, y = β(γ − α)2, z = γ(α− β)2, α+ β + γ = 0,

where (x, y, z) are barycentric coordinates, referred to an equilateral trian-gle. In what respect do its asymptotes behave differently from those of ahyperbola?

(b) Eliminate the parameters α, β, γ to obtain a single equation

x3 + y3 + z3 + a(x2y + xy2 + y2z + yz2 + z2x+ x2z) + bxyz = 0

for certain numbers a and b.(c) What equation does the curve have in polar coordinates?

Crux 786.S10.55*. (O.Bottema)

Let r − 1, r2, r3 be arbitrarily chosen positive numbers. Prove that thereexists a real triangle whose exradii are r1, r2, r3, and calculate the sides ofthis triangle.

Crux 787.S10.56*. (J.W.Lynch)

(a) Given two sides, a and b, of a triangle, what should be the length of thethird side, x, in order that the area enclosed be a maximum?

(b) Given three sides, a, b, c, of a quadrilateral, what should be thelength of the fourth side, x, in order that the area enclosed be a maximum?

See also Crux 914, on the interpretation of the positive root of

x3 − (a2 + b2 + c2)x− 2abc = 0.


Crux 790.S10.60. (R.H.Eddy)

Let ABC be a triangle with sides a, b, c in the usual order, and let a, b, cand ′a, ′b,

′c be two sets of concurrent cevians, with a, b, c intersecting

a, b, c in L, M , N respectively. If

a ∩ ′b = P, b ∩ ′c = Q, c ∩ ′a = R,

prove that, independently of the choice of concurrent cevians ′a, ′b, ′c, we

haveAP

PL· BQQM

· CRRN

=abc

BL · CM ·AN ≥ 8,

with equality occurring just when a, b, c are the medians of the triangle.

This problem extends Crux 588.


Given a triangle ABC, let ta, tb, tc be the lengths of its internal anglebisectors, and let Ta, Tb, Tc be the lengths of these bisectors extended to thecircumcircle of the triangle. Prove that

Ta + Tb + Tc ≥ 43(ta + tb + tc).

Solution. (B.Prielipp)

Ta + Tb + Tc ≥ 4s√3≥ 4

3√s(√s− a+

√s− b+

√s− c) ≥ 4

3(t1 + tb + tc).

The first inequality was established by Groenman in his solution to Crux628.S8.114, and the last two follow from item 8.9 of the Bottema Bible,where they are credited to Santalo (1943).

Crux 805.S10.120. (Klamkin)

If x, y, z > 0, prove that

x+ y + z

3≥ yz + zx+ xy√

y2 + yz + z2 +√z2 + zx+ x2 +

√x2 + xy + y2

,




Let LMN be the cevian triangle of the point S for triangle ABC. It istrivially true that S is the centroid of ABC =⇒ S is the centroid of LMN .Prove the converse.

Crux 808**.S10.132;11(1)18. (S.Rabinowitz)

Find the length o the largest circular arc ontained within the right trianglewith sides a ≤ b < c.


Let C be a given circle, and let Cim i = 1, 2, 3, 4, be circles such that(i) Ci is tangent to C at Ai, for i = 1, 2, 3, 4;(ii) Ci is tangent to Ci+1 for i = 1, 2, 3.Furthermore, let be a line tangnet to C at the other extremity of the

diameter of C through A1, and for i = 2, 3, 4, let A1Ai intersect at Pi.Prove that, if C, C1, and C4 are fixed, then the ratio of unsigned lengths

P2P3 : P3P4 is constant for all circles C2 and C3 that satisfy (i) and (ii).


Let D denote the point BC but by the internal hisector of angle BAC inthe Heronian triangle whose sides are c = 14, a = 13, b = 15. With D ascenter, describe the circle touching AC in L and cutting the extension ofAD in J . Show that AJ

AL =√

5+12 , the golden ratio.

Crux 815.S10.132*. (J.T.Groeman)

Prove the triangle inequalities

√3∑ 1

awa≥ 4sabc

,

3√

3∑ 1

awa∑awa

≥ 4

√2s

(abc)3,



Crux 816 (Tsintsifas)

Prove the triangle inequalities∏(b+ c) ≤ 8R(R+ 2r)s,∑bc(b+ c) ≤ 8R(R + r)s,∑a3 ≤ 8s(R2 − r2).

Crux 818.S10.159*. (A.P.Guinand)

Let ABC be a scalene triangle with P the point where the internal bisectorof A intersects the Euler line. If O,H,P are given, construct angle A, usingonly ruler and compass.


Of the two triangle inequalities

∑tan2 A

2≥ 1, and 2 − 8

∏sin

A

2≥ 1,

the first is well known and the second is equivalent to the the well knowninequality ∏

sinA

2≤ 1

8.

Prove or disprove the sharper inequality

∑tan2 A

2≥ 12 − 8

∏sin

A

2.


For i = 1, 2, 3, let Ai be the vertices of a triangle with opposite sides ai,let Bi be an arbitrary point on ai, and let Mi be the midpoint of BjBk.If the lines bi are perpendicular to ai through Bi, and if the lines mi areperpendicular to ai through Mi, prove that the bi are concurrent if and onlyif the mi are concurrent.


What are the coordinates of the intersection of the lines mi, given thatB1B2B3 is the pedal triangle of a point P?

Crux 829.S10.201. (Bottema)

Prove that the signed area of GIO is given by

−(a+ b+ c)(b − c)(c − a)(a− b)48 .


Let Rm be the circumradius of the trinagle formed by medians. Prove that

Rm ≥ a2 + b2 + c2

2(a+ b+ c).

See also Crux 733 and 970.

Crux 836.S10.228*,320*. (V.N.Murty)

(a)(1 − cosA)(1 − cosB)(1 − cosC) ≥ cosA cosB cosC,

with equality if and only if the triangle is equilateral.(b) Deduce Bottema’s triangle inequality

(1 + cos 2A)(1 + cos 2B)(1 + cos 2C) + cos 2A cos 2B cos 2C ≥ 0.

Crux 844.S10.264. (P.M.Gibson)

(a) A triangle A0B0C0 with centroid G0 is inscribed in a circle Γ with centerO. The lines A0G0, B0G0, C0G0 meet Γ again in A1, B1, C1, respectively,and G1 is the centroid of triangle A1B1C1. A triangle A2B2C2 with centroidG2 is obtained in the same way from A1B1C1, and the procedure is repeatedindefinitely, producing triangles with centroids G3, G4, . . . .

If gn = OGn, prove that the sequence g0, g1, g2, . . . is descreasing andconverges to zero.

(b) Prove or disprove that a result similar to (a) holds for a tetrahedroninscribed in a sphere, or more generally, for an n−simplex inscribed in ann−sphere.


Crux 845.S10.266*. (B.M.Saler)

Let r1, r2, r3 be the focal radii (all from the same focus F) of the pointsP1, P2, P3 on the ellipse x2

a2 + y2

b2 = 1. A circle center F and radius r =3√r1r2r3 intersects the focal radii in P ′

1, P′2, P

′3 respectively. Find the ratio

of the areas of P1P2P3 and P ′1P

′2P

′3. (This is Theorema Elegantissimum

from Acta Eruditorum, A.D. 1771, p.131, author unknown.)


Prove the triangle inequalities

mambmc

m2a +m2

b +m2c

≥ r,

12Rmambmc ≥∑

a(b+ c)m2a,

4R∑

ama ≥∑

bc(b+ c),

2R∑ 1

bc≥∑ ma

mbmc.

Crux 850.S10.271. (V.N.Murty)

Let x = rR and y = s

R . Prove that

y ≥ √x(√

6 +√

2 − x),



Given are three distinct points A,B,C on a circle. A point P in the planehas the property that if the lines PA, PB, PC meet the circle again in A′,B′, C ′ respectively, then A′B′ = A′C ′. Find the locus of P .

Crux 856.S10.303*. (Garfunkel)

For a triangle ABC, let M = R−2r2R . An inequality P ≥ Q involving elements

of the triangle will be called strong or weak respectively, according as

P −Q ≤M or P −Q ≥M.

(a) Prove that the inequality

∑sin2 α

2≥ 3

4


is strong.(b) Prove that the inequality

∑cos2 α

2≥∑

sin β sin γ

is weak.


For n = 0, 1, 2, . . ., let Pn be a point in the plane whose distances from thesides satisfy

da : db : dc =1an

:1bn

:1cn.

(a) A point Pn being given, show how to construct Pn+2.(b) Using (a), or otherwise, show how to construct the point Pn for an

arbitrary given value of n.

Crux 859.S10.307;11(6)191. (V.N.Murty)

Let ABC be a triangle of type II, namely, α ≤ β ≤ π3 ≤ γ. It is known that

for such a trinagle x := rR ≥ 1

4 . Prove the stronger inequality

x ≥√

3 − 12

.


P is an interior point of ABC, Lines through P parallel to the sides ofthe triangle meet those sides in the points A1, A2, B1, B2, C1, C2 as shownin the figure. Prove that

(a) A1B1C1 ≤ 13ABC,

(b) A1C2B1A2C1B2 ≥ 23ABC.

Crux 865.S10.325*. (C.Kimberling)

Let x, y, z be the distances from the sides to a variable point P inside atriangle. Prove that if 0 = t = 1, the critical point of xt + yt + zt satifies

x : y : z = ap : bp : cp

where p = 1t−1 . Discuss limiting cases.


Crux 866.S10.327*. (Jordi Dou)

Given a triangle ABC with sdies a, b, c, find the minimum value of a ·XA+b · XB + c · XC, where X ranges over all the points of the plane of thetriangle.

Crux 872(corrected 10.18)S10.334. (G.Tsintsifas)

Let P be a point other than a vertex in the plane of a triangle ABC. It isknown that there exists a triangle with sides a ·PA, b ·PB, c · PC. If R0 isthe circumradius of this triangle, prove that

PA · PB · PC ≤ RR0.

When does equality hold ?


Let Ka, Kb, Kc be the circles with centers A, B, C and radii λ√bc, λ

√ca,

λ√ab. Find the locus of the radical center of Ka, Kb, Kc as λ ≥ 0.

Crux 880**.S11(1)26*. (C.Kimberling)

For a given triangle ABC, what curve is formed by all the points P inthree-dimensional space satisfying

BPC = CPA = APB?

Crux 883.S11(1)28*. (J.Tabov and S.Troyanski)

Let ABC be a triangle with area S, sides a, b, c, medians ma, mb, mc, andinterior angle bisectors ta, tb, tc. If

ta ∩mb = P, tb ∩mc = Q, tc ∩ma = R,

prove that σS < 1

6 , where σ is the area of triangle PQR.

P =c ·A+ b ·B + c · C

b+ 2c, Q =

a ·A+ c · B + a · Cc+ 2a

, R =b ·A+ a ·B + b · C

a+ 2b.

Note that the coordinates of R on p.29 is misprinted. The ratio

[PQR][ABC]

=bc2 + ca2 + ab2 − 3abc(b+ 2c)(c + 2a)(a+ 2b)

.

See also Curx 588, 585, and 790.


Crux 890.S11(2)55. (L.F.Meyers)

Construct triangle ABC, with straightedge and compass, given the lengthsof b an c of two sides, the midpoint Ma of the third side, and the foot Ha ofthe altitude to that third side.

Crux 892.S11(2)57*. (Stan Wagon)

ABCD is a square and ECD an isosceles triangle with base angles 15, asshown in the figure. Prove that AEB = 60 (and therefore triangle AEBis equilateral).

This problem is very well known, but all published solutions use trigonom-etry and/or auxiliary lines. What is required is a simple proof with trigonom-etry or any auxiliary lines (or circles).

Proof. (K.S.Williams) Write AEB = 2x, and BEC = y, in degrees.Then, BAE = 90 − x, and

2x ≥ 60 ⇐⇒ x ≥ 30 ⇐⇒ BC ≤ BE ⇐⇒ AB ≤ BE ⇐⇒ 2x ≤ 90−x⇐⇒ 2x ≥ 60.

Therefore, AEB = 2x = 60.

Editor’s remark: Such a proof can actually be found in Coxeter andGreitzer, Geometry Revisited, pp.25, 158.

Crux 895.S11(2)60. (J.T.Groenman)

Let ABC be a triangle with sides a, b, c in the usual order and circumcircleΓ. A line through C meets the segment AB in D, Γ again in E, and theperpendicular bisector of AB in F . Assume that c = 3b.

(a) Construct the line for which the length of DE is maximal.


(b) If DE has maximal length, prove that DF = FE.(c) If DE has maximal length and also CD = DF , find a in terms of b

and the measure of angle A.

Crux 896.S11(2)62. (J.Garfunkel)

∑sin2 A

2≥ 1 − 1

4

∏cos

B − C

2≥ 3

4.

Crux 902.S11(3)86. (J.C.Fisher)

(a) For any point P on a side of a given triangle, define Q to be that pointon the triangle for which PQ bisects the area. What is the locus of themidpoint of PQ?

(b) Like the curve in part (a), the locus of the midpoints of the perimeter- bisecting chords of a triangle (see Crux 674) has an orientation that isopposite to that of the given triangle. Is this a general principle? Moreprecisely, given a triangle and a family of chords joining P (t) to Q(t), where(i) P (t) and Q(t) move counterclockwise about the triangle as t increasesand (ii) P (t) = Q(t) for any t, does the midpoint of PQ always trace a curvethat is clockwise oriented?

Crux 903.S11(3)88. (S.Rabinowitz)

Let ABC be an acute - angled triangle with circumcenter O and orthocenterH.

(a) Prove that an ellipse with foci O and H can be inscribed in thetriangle.

(b) Show how to construct, with straightedge and compass, the points L,M , N where this ellipse is tangent to the sides BC, CA, AB of the triangle.

(c) Prove that AL, BM , CN are concurrent.

Crux 904.S11(3)90. (G.Tsintsifas)

Let M be any point in the plane of triangle ABC. The cevians AM , BM ,CM intersect the lines BC, CA, AB in A′, B′, C ′ respectively. Find thelocus of the point M such that

[MCB′] + [MAC ′] + [MBA′] = [MC ′B] + [MA′C] + [MB′A].



Let ABC be a triangle that is not right angled at B or C. Let D be thefoot of the perpendicular from A upon BC, and let M and N be the feet ofthe perpendiculars from D upon AB and AC, respectively.

(a) Prove that, if A = 90circ, then BMC = BNC.(b) Prove or disprove the converse of (a).

Crux 908.S11(3)93*. (M.S.Klamkin)


P = sinαA sinβ B sinγ C,

where A, B, C are the angles of a triangle, and α, β, γ are given positivenumbers.

Crux 910.S11(3)96*. (O.Bottema)

Determine the locus of the centers of the conics through the incenter andthe three excenters of a given triangle.

Pedoe pointed out that this appears in Durell’s Projective Geometry,p.203, and wrote

The incenter is the orthocenter of the triangle formed by thethree excenters, so that all conics through the four points arerectangular hyperbolas, and the locus of the centers of theseconics is the nine-point circle of the triangle formed by any threeof the four points, and this is the circumcircle of the originaltriangle.

Editor’s remark: the locus of the foci of all parabolas tangent to thethree sides of a triangle is the circumcircle of the triangle. (W.P. Milne,Homogeneous Coordinates, (1924) p.118).

Crux 919.S11(4)131. (J.Dou)

Show how to construct a point P which is the centroid of triangle A′B′C ′

where A′, B′, C ′ are the orthogonal projections of P upon the three givenlines a, b, c respectively.

Solution. The symmedian point is the only point which is the centroidof its own pedal triangle.


Problem

Find the point P which is the circumcenter of its cevian triangle.

Crux 920.S11(4)131. (B.C.Rennie)

If a triangle of unit base and unit altitude is in the unit square. Show thatthe base of the triangle must be one side of the triangle.

Crux 923.S11(5)150. (C.Kimberling)

Let the ordered triple (a, b, c) denote the triangle whose side lengths are a,b, c. Similarity being an equivalence relation on the set of all triangles, letthe ordered ratios a : b : c (which we call a triclass) denote the equivalenceclass of all triangles (a′, b′, c′) such that

a : a′ = b : b′ = c : c′.

Let T be the set of all triclass. A multiplication on T is defined by

a : b : cα : β : γ = aα+(c−b)(γ−β) : bβ+(a−c)(α−γ) : cγ+(b−a)(β−α).

(a) Prove that (T, ) is a group.(b) If T is the set of all a : b : c such that a, b, c are integers, prove that

every triclass in T is a unique product of “prime” triclasses.

Crux 925.S11(5)154*. (J.T.Groenman)

The points Ai, i = 1, 2, 3, are the vertices of a triangle with sides ai andmedian lines mi. Through a point P , the lines parallel to mi intersects ai

in Si. Find the locus of P if the three points Si, i = 1, 2, 3 are collinear.See also Crux 1073.Solution. Let P = xA+ yB + zC in barycentric coordinates. The point

X is P+k(−2A+B+C) for some k. This is (x−2k)A+(y+k)B+(z+k)C,and k = x

2 if this lies on BC. The point X is therefore (y+ x2 )B+(z+ x

2 )C.Similarly, Y and Z are the points Y = (x + y

2 )A + (z + y2 )C and Z =

(x+ z2)A+ (y + z

2 )B.These are collinear if and only if

det

⎛⎝ 0 x+ 2y x+ 2z

2x+ y 0 y + 2z2x+ z 2y + z 0

⎞⎠ = 0.


This reduces to (x + y + z)(xy + yz + zx) = 0. The locus is therefore theSteiner ellipse.

AX, BY , CZ are collinear if and only if (x− y)(y − z)(z − x) = 0.


Let P be a fixed point inside an ellipse, L a variable chord through P , andL′ the chord through P that is perpendicular to L. If P divides L intotwo segments of lengths m and n, and if P divides L′ into two segments oflengths r and s, prove that 1

mn + 1rs is a constant.

Crux 929.S11(5)159. (K.Satyanarayana)

Given a triangle ABC, find all interior points P such that, if AP , BP , CPmeet the circumcircle again in A1, B1, C1, then triangles ABC and A1B1C1

are congruent.


Crux 934.S11(6)194. (Leon Bankoff)

As shown in the figure, the diameter AB, a variable chord AJ , and theintercepted minor arc JB of a circle (O) form a mixtilinear triangle whoseinscribed circle (W ) touches arc JB in K and whose mixtilinear excircle (V )touches arc JB in L. The projections of W and V upon AB are C and Drespectively. As J moves along the circumference of circle (O), the ratio ofthe arcs KL and LB varies.

(a) When arcs KL and LB are equal, what are their values?(b) Show that BD is equal to the side of the inscribed square lying in

the right angle of triangle ADV .

Crux 937.S11(6)199. (Jordi Dou)

ABCD is a trapezoid in a cirlceφ, with AB//DC. The midpoint of AB isM , and the line DM meets the circle again in P . A line through P meetsthe lines BC in A′, CA in B′, AB in C ′, and the circle again in F ′. Provethat (A′B′, C ′F ′) is a harmonic range.


ABC is an acute triangle with AB < AC, and orthocenter H. M being aninterior point of segment DH, lines BM and CM intersect sides CA andAB in B′ and C ′ respectively. Prove that BB′ < CC ′.



∏sinB sinC ≤ 7

4+ 4

∏sin

A

2≤ 9

4.

Crux 947.S11(7)233. (Jordi Dou)

Let ABCD be a quadrilateral (not necessarily convex) with AB = BC,CD = DA, and AB ⊥ BC. The midpoint of CD being M , points K andL are found on line BC such that AK = AL = AM . If P , Q, R are themidpoints of BD, MK, ML respectively, prove that PQ ⊥ PR.

Crux 1025. (P. Messer)

A paper square ABCD is folded so that vertex C falls on AB and side CDis divided into two segments of lengths and m, as shown in the figure. Findthe minimum value of the ratio

m .

Crux 1038.S12.216. (J.Dou)

Given are two concentric circles and two lines through their centers. Con-struct a tangent to the inner circle such that one of its points of intersectionwith the outer circle is the midpoint of the segment of the tangent cut offby the two given lines.

Crux 1039.S13.152;14.176. (K.Satyanarayana)

Given are three collinear points O, P , H (in that order) such that OH <3OP . Construct a triangle ABC with circumcenter O and orthocenter Hand such that AP is the internal bisector of angle A. How many suchtriangles are possible?

Crux 1054 (P. Messer)

A paper square ABCD is divided into three strips of equal area by th parallellines PQ and RS. The square is then folded so that C falls on AB and Sfalls on PQ. Determine the ratio AC

CB .

Crux 1059.S12(10)290. (Kimberling)

In his book, The Modern Geometry of the Triangle, (London, 1913), W.Gallatlydenotes by J the circumcenter of triangle I1I2I3, whose vertices are the ex-centers of the reference triangle ABC. On pages 1 and 21 are figures in


which J appears to be collinear with the incenter and the circumcenter oftriangle ABC. Are these points really collinear?

Crux 1062.S13(1)17. (Klamkin)

(a) Let Q be a convex quadrilateral inscribed in a circle with center O.Prove:

(i) If the distance of any side of Q from O is half the length of theopposite side, then the diagonals of Q are orthogonal.

(ii) Conversely, if the diagonals of Q are orthogonal, then the distanceof any side of Q from O is half the length of the opposite side.

(b)∗ Suppose a convex quadrilateral Q inscribed in a centrosymmetricregion with center O satisfies either (i) or (ii). Prove or disprove that theregion must be a circle.


Triangles ABC and DEF are similar, with angles A = D, B = E, C = F ,and ratio of similarity λ = EF

BC . Triangle DEF is inscribed in triangle ABC,with D, E, F on the lines BC, CA, AB, not necessarily respectively. Threecases can be considered.

Case 1: D, E, F on BC, CA, AB respectively;Case 2: D, E, F on CA, AB, BC respectively;Case 3: D, E, F on AB, BC, CA respectively.For case 1, it is known that λ ≥ 1

2 [See Crux 606]. Prove that, for eachof cases 2 and 3, λ ≥ sinω, where ω is the Brocard angle of triangle ABC.

Crux 1073.S13(2)56. (J.Dou)

Let K be an interior point of triangle ABC. Through a point P in the planeof the triangle, parallels to the cevians AK, BK, CK are drawn to meetBC, CA, AB at L, M , N respectively. If the points L, M , N are collinear,

(a) prove that the locus of P is an ellipse;(b) construct the center of this ellipse;See also Crux 925, which is the special case when K = G.


Let ABC be a triangle with circumcenter O. Prove that(a) there are two points P in the plane of the triangle such that

PA2 : PB2 : PC2 = sec2A : sec2B : sec2C;


(b) these two points and O are collinear;(c) these two points are inverse with respect to the circumcircle of the

triangle.

Crux 1075.S13(2)60*. (J.Dou)

Let P be an interior point of triangle ABC. Denote by ρ and ρ′ the inradiiof triangle ABC and the pedal triangle of P . Prove that

OP ≥ OI ⇒ ρ′ ≤ 12ρ.

Give an example to show the converse does not hold. [Notation changedand statement corrected].

Crux 1076.S13(2)62. (M.S.Klamkin)

Let x, y, z denote the distances from an interior point of a given triangleABC to the respective vertices A, B, C; and let K be the area of the pedaltriangle of P with respect to ABC. Show that

x2 sin 2A+ y2 sin 2B + z2 sin 2C + 8K

is a constant independent of P .

Crux 1090.S13(4)126*. (Dan Sokolowsky)

Let Γ be a circle with center O, and A a fixed point distinct from O in theplane of Γ. If P is a variable point on Γ and AP meets Γ again in Q, findthe locus of the circumcenter of triangle POQ as P ranges over Γ.


Let A1A2A3 be a triangle and γi the excircle opposite Ai, i = 1, 2, 3. Apol-lonius knew how to construct the circle Γ internally tangent to the threeexcircles and encompassing them. Let Bi be the point of contact of Γ andγi, i = 1, 2, 3. prove that the lines A1B1, A2B2, A3B3 are concurrent.

Crux 1100.S13(5)160;(8)258. (D.J. Smeenk)

ABC is a triangle with C = 30, circumcenter O and incenter I. Points Dand E are chosen on BC and AC respectively, such that BD = AE = AB.Prove that DE = OI and DE ⊥ OI.

See also Crux 1196.


Crux 1103.S13(5)163. (R. Izard)

Three concurrent cevians through the vertices A, B, C of a triangle meetthe lines BC, CA, AB in D, E, F respectively, and the internal bisectorof angle A meets BC in V . If A, F , D, V , E are all concyclic, prove thatAD ⊥ BC.


The directly similar triangles ABC and DEC are both right angled at C.Prove that

(a) AD ⊥ BE;(b) AD

BE equals the ratio of similitude of the two triangles.Pedoe: The problem can be generalized (the triangles need not be right-

angled), but (b) is incorrectly stated. The generalized statement is:The directly similar triangles ABC and DEC have angle Ω at C. Prove

that(a) AD makes angle Ω with BE;(b) AD

BE = CACB = CD

CE .

Crux 1109.S13(10)322*;14(3)78. (D.J. Smeenk)

ABC is a triangle with orthocenter H. A rectangular hyperbola with centerH intersects line BC in A1 and A2, line CA in B1 and B2, and line AB inC1 and C2. Prove that the midpoints of A1A2, B1B2 and C1C2 are collinear.

Crux 1140.S13(6)232. (J.Dou)

Given triangle ABC, construct a circle which cuts (extended) lines BC, CA,AB in pairs of points A′ and A′′, B′ and B′′, C ′ and C ′′ respectively suchthat angles A′AA′′, B′BB′′ and C ′CC ′′ are all right angles.

Crux 1170.S13.332. (C.Kimberling)

In the plane of triangle ABC, let P and Q be points having trilinears α1 :β1 : γ1 and α2 : β2 : γ2, respectively, where at least one of the productsα1α2, β1β2, γ1γ2 is nonzero. Give a euclidean construction for the pointP ∗Q having trilinears α1α2 : β1β2 : γ1γ2. (A point has trilinears α : β : γ ifits signed distances to the sides BC, CA, AB are respectively proportionalto the numbers α, β, γ.)


Crux 1171.S13.333. (D.S.Mitrinovic and J.E.Pecaric)

(i) Determine all real numbers λ so that, whenever a, b, c are the lengthsof three segments which can form a triangle, the same is true for (b + c)λ,(c+ a)λ, (a+ b)λ. [For λ = −1, we have Crux 14.S1.28].

(ii) Determine all pairs of real numbers λ, µ so that, whenever a, b, c arethe lengths of three segments which can form a triangle, the same is true for(b+ c+ µa)λ, (c+ a+ µb)λ, (a+ b+ µc)λ.


Suppose ABC is an acute triangle. Prove that there is an point inside ABCand points D, E on BC, F , G on CA, and H, I on AB such that GPH,IPD, and EPF are congruent equilateral triangles.

Crux 1177.S14(1)20. (Tsintsifas)

ABC is a triangle and M an interior point with barycentric coordinates(λ1, λ2, λ3). Lines HMD, JMF , EMI are parallel to AB, BC, CArespectively. The centroids of triangles DME, FMH, IMJ are denoted byG1, G2, G3 respectively. Prove that

[G1G2G3] =13(λ1λ2 + λ2λ3 + λ3λ1)[ABC].

Crux 1180.S14(1)24**. (J.R.Pounder)

(a) It is well known that the Simson line of a point P on the circumcircle ofa triangle T envelopes a deltoid (Steiner’s hypocycloid) as P varies. Showthat this is true of an oblique Simson line as well. (An oblique Simson lineof T is the line passing through the points A1, B1, C1 chosen on edges BC,CA, AB respectively so that the lines PA1, PB1, PC1 make equal angles(say θ) in the same sense of rotation, with BC, CA, AB respectively. Theusual Simson line occurs when θ = 90circ.

(b*) Given such an oblique deltoid for T , locate a triangle T ′ similarto T such that the “normal” deltoid for T ′ and the oblique deltoid for Tcoincide.


Let ABC be a nonequilateral triangle and let O, I, H, F denote the cir-cumcenter, incenter, orthocenter, and the centero fo the nine-point circle,respectively. Can either of the triangles OIF or IFH be equilateral?


Crux 1188.S14(1)32. (D.Sokolowsky)

Given a circle K and distinct points A, B in the plane of K, construct achord PQ of K such that B lies on the line PQ and PAQ = 90.

Crux 1191.S14(2)55. (H.Fukagawa)

Let ABC be a triangle, and let points D, E, F be on sides BC, CA, ABrespectively such that triangles AEF , BFD, and CDE all have the sameinradius r. Let r1 and r2 denote the inradii of DEF and ABC respectively.Show that r + r1 = r2.

Crux 1192.S14(2)56. (R.K.Guy)

Let ABC be an equilateral triangle and v, w be arbitrary positive realnumbers. S (respectively T , U) is the Apollonius circle which is the locusof points whose distances from A and B (respectively A and C, B and C)are in the ratio v : w (respectively v : v + w, w : v + w). Prove that S,T , U have just one point in common, and that it lies on the circumcircle oftriangle ABC.


Let ABC be a triangle with medians ma, mb, mc and circumcircle Γ. LetDEF be the triangle formed by the parallels to BC, CA, AB through A,B, C respectively, and let Γ′ be the circumcircle of DEF . Let A′, B′, C ′

be the triangle formed by the tangents to Γ at the points (other than A, B,C) where ma, mb, mc meet Γ. Finally let A′′, B′′, C ′′ be the points (otherthan D, E, F ) where ma, mb, mc meet Γ′. prove that the lines A′A′′, B′B′′,C ′C ′′ concur in a position on the Euler line of triangle ABC.

Crux 1196.S14(2)62. (J.Dou)

Let I be the incenter and O the circumcenter of triangle ABC. Let Don AC and E on BC be such that AD = BE = AB. Prove that DE isperpendicular to OI.

See also Crux 1100.

Crux 1198.S14(3)85,(6)179,(10)318. (Groenman)

Let ABC be a triangle with incenter I, Gergonne point G, and Nagel pointN . Let J be the isotomic conjugate of I. Prove that G, N , J are collinear.


Crux 1203.S14.91. (M.N.Naydenov)

A quadrilateral inscribed in a circle of radius R and circumscribed arounda circle of radius r has consecutive sides a, b, c, d, semiperimeter s and areaF . Prove that

(a) 2√F ≤ s ≤ r =

√r2 + 4R2;

(b) 6F ≤ ab+ ac+ ad+ bc+ bd+ cd ≤ 4r2 + 4R2 + 4r√r2 + 4R2;

(c) 2sr2 ≤ abc+ abd+ acd+ bcd ≤ 2r[r +√r2 + 4R2]2;

(d) 4Fr2 ≤ abcd ≤ 169 r

2(r2 + 4R2).

Crux 1216.S14.120;21.131. (W.Janous)


2 <sinAA

+sinBB

+sinCC

≤ 9√

32π

.

Crux 1217.S14.123;20.297. (N.Bejlegaard)

Given are two lines 1 and 2 intersecting at A, and a point P in the sameplane, where P does not lie on either angle bisector at A. Also given is apositive real number r.

(a) Construct a line through P , intersecting 1 and 2 at B and C re-spectively, such that AB +AC = r.

(b) Construct a line through P , intersecting 1 and 2 at B and C re-spectively, such that |AB −AC| = r.

Crux 1219.S14.124. (H.Freitag and D.Sokolowsky)

Let the incircle of triangle ABC touch AB at D, and let E be a point ofside AC. Prove that the incircles of triangles ADE, BCE and BDE havea common tangent.

Crux 1224.S14(5)145,(8)236. (Tsintsifas)

A1A2A3 is a triangle with circumcircle Ω. Let x1 < X1 be the radii of thecircles tangent to A1A2, A1A3, and arc A2A3 of Ω. Let x2 and X2 be definedsimilarly. Prove that

(a) x1X1

+ x2X2

+ x3X3

= 1;(b) X1 +X2 +X3 ≥ 3(x1 + x2 + x3) ≥ 12r, where r is the inradius of

triangle A1A2A3.



Find all points whose pedal triangles with respect to a given triangle areisosceles and right-angled.


Let ABC be a triangle and M an interior point with barycentric coordinates(λ1, λ2, λ3). The distances of M from the vertices A, B, C are x1, x2, x3,and the circumradii of triangles MBC, MCA, MAB, ABC are R1, R2, R3,R. Show that

λ1R1 + λ2R2 + λ3R3 ≥ R ≤ λ1x1 + λ2x2 + λ3x3.

Crux 1252.S14(7)211*. (Tsintsifas)

Let ABC be a triangle and M an interior point with barycentric coordinatesλ1, λ2, λ3. We denote the pedal triangle and the cevian triangle of M byDEF and A′B′C ′ respectively. Prove that

[DEF ][A′B′C ′]

≥ 4λ1λ2λ3(s

R)2,

where s is the semiperimeter and R the circumradius of triangle ABC.

Crux 1260.S14(8)236*;15(2)51. (H.Fukagawa)

Let ABC be a triangle with angles B and C acute, and let H be the footof the perpendicular from A to BC. Let O1 be the circle intrnally tangentto the circumcircle O of triangle ABC and touching the segments AH andBH. Let O3 be the circle similarly tangnet to O, AH, and CH. Finally, letO2 be the incircle of triangle ABC, and denote the radii of O1, O2, O3 byr1, r2, r3 respectively. Show that

(a) r2 = r1+r32 ;

(b) the centero of O1, O2, O3 are collinear.

Crux 1269.S14(9)270. (Janous)

Let ABC be a non-obtuse triangle with circumcenter M and circumradiusR. Let u1, u2, u3 be the lengths of the parts of the cevians (through M)between M and the sides opposite to A, B, C respectively. Prove or disprovethat

R

2≤ u1 + u2 + u3

3< R.


Crux 1272.S14(8)256. (J.T. Groenman)

Let A1A2A3 be a triangle. Let the incircle have center I and radius ρ, andmeet the sides of the triangle at points P1, P2, P3. Let I1, I2, I3 be theexcenters and ρ1, ρ2, ρ3 the exradii. Prove that

(a) the lines I1P1, I2P2, I3P3 concur at a point S;(b) the distances d1, d2, d3 of S to the sides of the triangle satisfy

d1 : d2 : d3 = ρ1 : ρ2 : ρ3.

This is the point X57 = as−a : b

s−b : cs−c .


Let ABC be a triangle, M an interior point, and A′B′C ′ its pedal triangle.Denote the sides of the two triangles by a, b, c, and a′, b′, c′ respectively.Prove that

a′

a+b′

b+c′

c< 2.

Crux 1275.S14(9)279. (P.Penning)

On a circle C with radius R three points A1, A2, A3 are chosen arbitrarily.Prove that the three circles with radiusR, not coinciding with C, and passingthrough two of the points A1, A2, A3 intersect in the orthocenter of triangleA1A2A3.

Crux 1279.S14(9)284*. (Dou)

Consider a triangle whose orthocenter lies on its incircle.(a) Show that if one of its angles is given, the others are determined.(b) Show that if it is isosceles, then its sides are in the proportion 4:3:3.

Crux 1280.S14(9)287. (Janous)

Let ABC be a triangle and let A1, B1, C1 be points on BC, CA, ABrespectively, such that

A1C

BA1=B1A

CB1=C1B

AC1k > 1.

Show thatk2 − k + 1k(k + 1)

<perimeter (A1B1C1)perimeter (ABC)

<k

k + 1,

and that both bounds are best possible.



Let ABC be a triangle, I the incenter, and A′, B′, C ′ the intersections ofAI, BI, CI with the circumcircle. Show that

IA′ + IB′ + IC ′ − (IA+ IB + IC) ≤ 2(R − 2r).

Crux 1290.S14(10)314. (J.B.Tabov)

The triangles B1B2B3 and C1C2C3 are homothetic and each of them is inperspective with the triangle A1A2A3 (vertices with the same index corre-spond). Di (i = 1, 2, 3) is the midpoint of the segment BiCi. Prove thattriangle A1A2A3 and D1D2D3 are in perspective.

Comment by Jordi Dou: The proposition is false!

Crux 1293.S15(1)16. (S.Mauer)

Solve the following “twin” problems. In both cases, O is the center of thecircle.

(a) In Figure (a), AB = BC and ABC = 60. Prove that CD = OA√

3.(b) In Figure (b), OA = BC and ABC = 30. Prove that CD =

AB√

3.


Let A1A2A3 be a triangle with I1, I2, I3 the excenters and B1, B2, B3 thefeet of the altitudes. Show that the lines I1B1, I2B2, I3B3 concur at a pointcollinear with the incenter and circumcenter of the triangle.

This is the point X46 = [a(a3 + a2(b+ c)− a(b2 + c2)− (b+ c)(b− c)2)].Kimberling notes that this is a known result (Casey, Analytic Geometry,

2nd ed., Hodges & Figgis, Dublin, 1893, p.85).


Let A1A2A3 be an acute triangle with circumcenter O. Let P1, Q1 denotethe intersection of A1O with A2A3 and with the circumcircle respectively.Define P2, Q2, P3, Q3 analogously. Prove that

(a) OP1P1Q1

· OP2P2Q2

· OP3P3Q3

≥ 1;(b) OP1

P1Q1+ OP2

P2Q2+ OP3

P3Q3≥ 3;

(c) A1P1P1Q1

· A2P2P2Q2

· A3P3P3Q3

≥ 27.


Crux 1307.S15(2)58. (Dou)

Let A′, B′, C ′ be the intersections of the bisectors of triangle ABC with theopposite sides, and let A′′, B′′, C ′′ be the midpoints of B′C ′, C ′A′, A′B′

respectively. Prove that AA′′, BB′′, CC ′′ are concurrent.


Let ABC be a triangle with circumcircle Γ, and let DEF be the triangleformed by the lines tangent to Γ at A, B, C. Call a triangle A′B′C ′ acircumcevian triangle if for some point P , A′ is the point other than A whereAP meets Γ, and similarly for B′ and C ′. Prove that DEF is perspectivewith every circumcevian triangle.


Let ABC be a triangle with medians AD, BE, CF and median point G.We denote triangles AGF , BGF , BGD, CGD, CGE, AGE by i, i =1, 2, 3, 4, 5, 6 respectively. LetRi and ri denote the circumradius and inradiusof i. Prove that

(i) R1R3R5 = R2R4R6;(ii) 15

2r <1r1

+ 1r3

+ 1r5

= 1r2

+ 1r4

+ 1r6< 9

r .

Crux 1317.S15(3)91;16(3)80. (A.Bondesen)

Crux 1133 suggests the following problem. In a triangle ABC the excircletouching side AB touches line BC and AC at points D and E respectively.If AD = BE, must the triangle be isosceles?

Crux 1321.S15(4)116*;16(3)81. (Dou)

The circumcircle of a triangle is orthogonal to an excircle. Find the ratio oftheir radii.

Answer: R : rc = 1 : 2.

Crux 1329.S15(4)126*. (D.J. Smeenk)

Let ABC be a triangle, and let congruent circles C1, C2, C3 be tangent tohalf lines AB and AC, BA and BC, CA and CB, respectively.

(a) Determine the locus of the circumcenter P of triangle DEF , whereD, E, F are the centers C1, C2, C3.


(b) If C1, C2, C3 all pass through the same point, show that

1ρ

=1r

+1R,

where r and R are the inradius and the circumradius of triangle ABC.

Crux 1342.S15(6)188*. (Groenman)

Let ABC be a triangle and let D and E be the midpoints of BC and ACrespectively. Suppose that DE is tangent to the incircle of triangle ABC.Prove that rc = r, where r is the inradius of triangle ABC and rc theexradius to AB.

Crux 1343.S15(6)189*. (D.J. Smeenk)

ABC is an acute triangle and D, E are the feet of the altitudes to BC, ACrespectively. Suppose DE is tangent to the incircle. Show that rc = 2R,where R is the circumradius and rc is the exradius to AB.

Crux 1346.S15(6)190*. (D.J. Smeenk)

Let ABC be an isosceles triangle with AB = AC and A = 12. Let Don AC and E on AB be such that CBD = 42 and = 18. Find angleEDB.

Crux 1355.S15(8)240;16(3)81. (Tsintsifas)

Let ABC be a triangle and I its incenter. The perpendicular to AI at Iintersects the line BC at the point A′. Analogously, we define B′ and C ′.Prove that A′, B′, C ′ lie in a straight line.

Crux 1359.S15(8)244. (G.R. Veldkamp)

Let PQR, PST , and PUV be congruent isosceles triangles with commonapex P and having no vertex in common other than P . The sense P → Q→R, P → S → T , and P → U → V is anticlockwise. We suppose moreoverthat V Q and RS meet in A, RS and TU meet in B, and TU and V Q inC. Prove that P is on the line joining the circumcenter to the symmedianpoint of triangle ABC.


Crux 1372.S15(9)284*. (D.J. Smeenk)

Triangle ABC has circumcenter O and median point G, and the line AGand BG intersect the circumcircle again at A1 and B1 respectively. Supposethat A, B, O, G are concyclic. Show that

(a) AA1 = BB1;(b) triangle ABC is acute angled.

Crux 1376.S15(9)287. (Veldkamp)

Let ABCD be a quadrilateral with an inscribed circle of radius r and acircumscribed circle of radius R. Let AC = p and BD = q be the diagonals.Prove that

pq

4r2− 4R2

pq= 1.

Crux 1379.S15(10)307. (P.Penning)

Given are an arbitrary triangle ABC and an arbitrary interior point P . Thepedal-points of P on BC, CA, AB are D, E, F respectively. Show that htenormal from A to EF , from B to FD, and from C to DE are concurrent.

Crux 1385.S16(1)20. (Klamkin)

Show that the sides of the pedal triangle of any interior point of an equilat-eral triangle T are proportional to the distances from P to the correspondingvertices of T .


Let ABC be a triangle and D the point on BC so that the incircle of triangleABD and the excircle (to side DC) of triangle ADC have the same radiusρ1. Define ρ2 and ρ3 analogously. Prove that

ρ1 + ρ2 + ρ3 ≥ 94r,

where ρ is the inradius of triangle ABC.


Let A1A2A3 be a triangle with incenter I, excenters IA, IB , IC , and medianpoint G. Let H1 be the orthocenter of triangle I1A2A3, and define H2 and


H3 analogously. Prove that A1H1, A2H2, A3H3 are concurrent at a pointcollinear with G and I.

Crux 1395.S16(2)46,(10)299. (Janous)

Given an equilateral triangle ABC, find all points P in the same plane suchthat PA2, PB2, PC2 form a triangle.

Crux 1404.S16(3)83. (J.T.Groenman and D.J. Smeenk)

Let ABC be a triangle with circumradius R and inradius ρ. A theoremof Poncelet states that there is an infinity of triangles having the samecircumcircle and the same incircle as triangle ABC.

(a) Show that the orthocenters of these triangles lie on a circle.(b) If R = 4ρ, what can be said about the locus of the centers of the

nine-point circles of these triangles?


ABC is a triangle with sides a, b, c. The excircle to the side a has center Iaand touches the sides at D, E, F . M is the midpoint of BC.

(a) Show that the lines IaD, EF , and AM are concurrent at a point Sa.


(b) In the same way we have points Sb and Sc. Prove that

[SaSbSc] >32[ABC].

Crux 1425.S16(5)146;17(6)175. (Dou)

Let D be the midpoint of side BC of the equilateral triangle ABC and ωa circle through D tangnet to AB, cutting AC in points B1 and B2. Provethat the two circles, distinct from ω, which pass through D and are tangnetto AB, and which respectively pass through B1 and B2, have a point incommon on AC.

Crux 1430.S16(5)158*;17(2)48*. (M.Bencze)

AD, BE, CF are (not necessarily concurrent) cevians in triangle ABC,intersecting the circumcircle of triangle ABC in the pointsP , Q, R. Prove


thatAD

DP+BE

EQ+CF

FR≥ 9.

When does equality hold?Same as Math. Mag. 1402 (June 1993).

Crux 1432.S16(6)180*;17(1)18. (J.T.Groenman)

If the Nagel point of a triangle lies on the incircle, prove that the sum oftwo of the sides of the triangle equals three times the third side.

Crux 1436.S16(6)186*. (D.J. Smeenk)

A point P lies on the circumcircle Γ of a triangle ABC, P not coincidingwith one of the vertices. Circles Γ1 and Γ2 pass through P and are tangentto AB at B, and to AC at C respectively. Γ1 and Γ2 intersect at P and atQ.

(a) Show that Q lies on the line BC.(b) Show that as P varies over Γ the line PQ passes through a fixed

point on Γ.


Let A′B′C ′ be an equilateral triangle inscribed in a triangle ABC, so thatA′ ∈ BC, B′ ∈ CA, C ′ ∈ AB. If

BA′

A′C=CB′

B′A=AC ′

C ′B,

prove that triangle ABC is equilateral.


Let ABC be a triangle. If P is a point on the circumcircle, and D, E, F arethe feet of the perpendiculars from P to BC, AC, AB respectively, then it iswell known that D, E, F are collinear. Find P such that E is the midpointof the segment DF .

Crux 1444.S16(7)214. (Dou)

Given the center O of a conic γ and three pointsA, B, C lying on γ, constructthose points X on γ such that XB is the bisector (interior or exterior) ofangle AXC.



Let A′B′C ′ be an equilateral triangle inscribed in triangle ABC, so thatA′ ∈ BC, etc. Denote by G′, G the centroid, by O′, O the circumcenters,by I ′, I the incenters, and by H ′, H the orthocenters of triangles A′B′C ′

and ABC respectively. Prove that in each of the four cases(a) G = G′,(b) O = O′,(c) I = I ′, [not correct; Seimiya gave a counterexample].(d) H = H ′,ABC must be equilateral.

Crux 1453.S16(8)247. (D.J. Smeenk)

Triangle ABC moves in such a way that AB passes through a fixed point Pand AC passes through a fixed point Q. Prove that throughout the motion,BC is tangent to a fixed circle.


Let A′B′C ′ be a triangle inscribed in triangle ABC, so that A′ ∈ BC, etc.Suppose that

BA′

A′C=CB′

B′A=AC ′

C ′B= 1,

and that triangle A′B′C ′ is similar to triangle ABC. Prove that the trianglesare equilateral.


Let A′B′C ′ be a triangle inscribed in triangle ABC, so that A′ ∈ BC, etc.(a) Prove that

BA′

A′C=CB′

B′A=AC ′

C ′Bif and only if the centroids G, G′ of the two triangles coincide.

(b) Prove that if (1) holds, and either the circumcenters O, O′ or the or-thocenters H, H ′ of the triangles coincide, then triangle ABC is equilateral.

(C)* If (1) holds and the incenters I and I ′ of the triangle coincide,characterize triangle ABC.


Crux 1466.S16(9)285. (J.T.Groenman and D.J. Smeenk)

On the sides of triangle A1A2A3 and outside the triangle we draw similartriangles A3A2B1, A1A3B2 and A2A1B3 with geocenters G1, G2, and G3

respectively. The geocenters of triangles A1B3B2, A2B1B3, A3B2B1 andA1A2A3 are Γ1, Γ2, Γ3, and G respectively. It is known that G is thegeocenter of triangle B1B2B3 as well. [See Math. Mag. 50 (1985) 84 – 89].Show that Γ1G1 has midpoint G, length 2

3 |A1B1|, and is parallel to A1B1.


Let A′B′C ′ be an equilateral triangle inscribed in a triangle ABC, so thatA′ ∈ BC, B′ ∈ CA, C ′ ∈ AB, and so that A′B′C ′ and ABC are directlysimilar. If BA′ = CB′ = AC ′, prove that the triangle are equilateral. [Notcorrect!]

Crux 1476.S16(10)309. (K.R.S.Sastry)

A triangle is called self - altitude if it is similar to the triangle formed fromits altitudes. Suppose triangle ABC is self - altitude, with sides a ≥ b ≥ cand angles bisectors AP , BQ, CR. Prove that the lengths of CP , PB, BR,RA form a geometric progression.

Crux 1480.S16(10)316. (J.B.Romero Marquez)

ABC andA′B′C ′ are triangles connected by a dilatation (BC//B′C ′, CA//C ′A′,AB//A′B′) and A′′ = BC ′∩B′C, B′′ = AC ′∩A′C, C ′′ = AB′∩A′B). Showthat triangle A′′B′′C ′′ is connected to either of the two given triangles by adilatation, and that the centroids of the three triangles are collinear.

Crux 1483.S17(1)22*. (Tsintsifas)

Let A′B′C ′ be an equilateral triangle inscribed in a triangle ABC, so thatA′ ∈ BC, B′ ∈ CA, C ′ ∈ AB, and so that A′B′C ′ and ABC are directlysimilar.

(a) Show that, if the centroids G and G′ of the triangles coincide, theneither the triangles are equilateral or A′, B′,C ′ are the midpoints of the sidesof triangle ABC.

(b) Show that if either the circumcenters O, O′ or the incenters I, I ′ ofthe triangles coincide, then the triangles are equilateral.


Crux 1486.S17(2)50. (Dou)

Given three triangles T1, T2, T3 and three points P1, P2, P3, constructpoints X1, X2, X3 such that the triangles X2X3P1, X3X1P2 and X1X2P3

are directly similar to T1, T2, T3 respectively.


In triangle ABC, the internal bisector of angle A meet BC at D, and theexternal bisector of angles B and C meet AC and AB (produced) at E andF respectively. Suppose that the normals to BC, CA, AB at D, E, F ,respectively, meet. Prove that AB = AC.

Crux 1492.S17(2)50*;25(8)508*. (Tsintsifas)

Let A′B′C ′ be an equilateral triangle inscribed in a triangle ABC, so thatA′ ∈ BC, B′ ∈ CA, C ′ ∈ AB. Suppose also that BA′ = CB′ = AC ′.

(a) If either the centroids G, G′ or the circumcenters O, O′ of the trian-gles coincide, prove that triangle ABC is equilateral.

(b)* If either the incenters I, I ′ or the orthocenters H, H ′ of the trianglescoincide, characterize triangle ABC.

Crux 1500.S17(2)60 Sastry

A parallelogram is called self-daigonal if its sides are proportional to itsdiagonals. Suppose that ABCD is a self-diagonal parallelogram in whichthe bisectors of angles ADB meets ABat E. Prove that AE = AC −AB.

Crux 1510.S17.91;19.50,204;21.159. (J.Garfunkel)

Let P be any point inside triangle ABC. Line PA, PB, PC are drawn andangles PAC, PBA, PCB are denoted by α, β, γ respectively. Prove ordisprove that

cotα+ cot β + cot γ ≥ cotA

2+ cot

B

2+ cot

C

2,

with equality when P is the incenter of triangle ABC.

Crux 1573. (Seimiya)

Let M be the midpoint of BC of triangle ABC. Suppose that BAM = Cand MAC = 15. Find angle C.


Crux 1637.S18.125;20.165;24(7)427. (G.Tsintsifas)

Prove that ∑ sinB + sinCA

>12π

for a non-obtuse triangle.

Crux 1708.

ABC is a triangle of area 1, and B1, B2, C1, C2 are the points of trisectionof AB and AC respectively. Find the area of the quadrilateral formed bythe four lines CB1, CB2, BC1, BC2.

Crux 1722.S19(2)56. (Seimiya)

ABCD is a cyclic quadrilateral with BD < AC. Let E and F be theintersections of AB, CD and of BC, AD, respectively, and let L and M bethe midpoints of AC and BD. Prove that

LM

EF=

12

(AC

BD− BD

AC

).

Crux 1730.S19.81;20.18. (G.Tsintsifas)

Prove that in ABC,

∑bc(s− a)2 ≥ 1

2sabc.

Crux 1740.S19.94,305;21.159*. (Dan Pedoe)

In triangle ABC the points N , L, M , in that order on AC, are respectivelythe foot of the perpendicular from B to AC, the intersection with AC of thebisector of angle ABC, and the midpoint of AC. The nagles ABN , NBL,LBM and MBC are all equal. Determine the angles of triangle ABC.

Crux 1756.S19.6.172. (K.R.S.Sastry)

For positive integers n ≥ 3 and r ≥ 1, the n−gonal number of rank r isdefined as

P (n, r) = (n− 2)r2

2− (n− 4)

r

2.


Call a triple (a, b, c) of natural numbers, with a ≤ b ≤ c, an n−gonalPythagorean triple if P (n, a) + P (n, b) = P (n, c). When n = 4, we getthe usual Pythagorean triple.

(i) Find an n−gonal Pythagorean triple for each n.(ii) Consider all triangles ABC whose sides are n−gonal Pythagorean

triples for some n ≥ 3. Find the maximum and minimum possible values ofangle C.

Remark: See also S. Hirose, Fibonacci Quarterly, 24 (1986) 99 – 106.

Crux 1812.S20.20. (T.Seimiya)

ABC is a right triangle with right angle at C. Let D be ap oint on side AB,and let M be the midpoint of CD. Suppose that AMD = BMD. Provethat

ACD : BCD = CDA : CDB.

Crux 1814.S20.23*. (D.J. Smeenk)

Given are the fixed line l with two fixed points A and B on it, and a fixedangle ϕ. Determine the locus of the point C with the following property:the angle between l and the Euler line of ABC is equal to ϕ.

Crux 1820.S20.29. (J.B.Romero Margque)

Let O be the point of intersection of the diagonals AC and BD of thequadrangle ABCD. Prove that the orthocenters of the four triangles OAB,OBC, OCD, ODA are the vertices of a parallelogram that is similar to thefigure formed by the centroids of these four triangles. What if ‘centroids’ isreplaced by ‘circumcenter’ ?

Crux 1827.S20.57;21.54*;22.36*,78. (S.Arslanagoc and M.M.Milssevi’c)

(i)∑ bc

A(s−a) ≥ 12sπ .

(ii) It follows easily from the proof of Crux 1611 and the correction on19.79 that also ∑ b+ c

A≥ 12s

π.

Do the two summations above compare in general?


Crux 1843.S20.113;21.55*. (S.Arslanagoc and M.M.Milssevi’c)

(i)∑ a

2A(s−a) ≥ 9π .

(ii) It is obvious that ∑ 1A

≥ 9π.

Do these two summations compare in general?

Crux 1880. (Sastry)

AD and BE are angle bisectors of triangle ABC, with D on BC and E onAC. Suppose AD = AB and BE = BC. Determine the angles of triangleABC.

Crux 1895.S20.263;21.204*. (J.Chen and G.Yu)

Let P be an interior point of triangle A1A2A3; R1, R2, R3 the distancesfrom P to A1, A2, A3; and R the circumradius of triangle A1A2A3. Provethat

R1R2R3 ≤ 3227R3,

with equality when A2 = A3 and PA2 = 2PA1.

Crux 1897 (Sastry)

In triangle ABC the bisectors of angles B and C meet the median AD atE and F respectively. If BE = CF , prove that triangle ABC is isosceles.

Crux 1902.S (Kuczma)

ABC is a triangle with circumcircle Γ. Let P be a variable point on the arcACB of Γ, other than A, B, C. X and Y are points on the rays AP abdBP respectively such that AX = AC and BY = BC. Prove that the lineXY always passes through a fixed point.

See also Crux 1993.

Crux 1904.S21.204*. (K.W.Lau)

Prove that

ma(bc− a2) +mb(ca− b2) +mc(ab− c2) ≥ 0.


Crux 1906.S (K.R.S.Sastry)

Let AP bisect angle A of triangle ABC, with P on BC. Let Q be the pointon segment BC such that BQ = CP . Prove that

AQ2 = AP 2 + (b− c)2.

Crux 1908.S20(10)293. (C.J.Bradley)

In ABC the feet of the perpendiculars from A, B, C onto BC, CA, ABare denoted by D, E, F respectively. H is the orthocenter. The triangle issuch that all of AH −HD, BH −HE and CH −HF are positive. K is aninteral point of ABC and L, M , N are the feet of the perpendiculars fromK onto BC, CA, AB respectively. Prove tht AL, BM , CN are concurrentif KL : KM : KN is equal to

(i) AH −HD : BH −HE : CH −HF ;(ii) 1

AH−HD : 1BH−HE : 1

CH−HF .


1995 – 1999

Crux 1910.S21.22 (J.Kotani)

The octahedron ABCDEF is inscribed in a sphere so that the three diag-onals AF , BD, CE meet at a point, and the centroids of the six triangularfaces of the octahedron are also inscribed in a sphere. Show that

(i) the orthocneters of the six faces are inscribed in a sphere;(ii) (AB ·DF = AD ·BF )(AC ·EF +AE ·CF )(BC ·DE+CD ·BE) =

36V 2,where V is the volume of the octahedron.


ABC is a triangle with AB = AC. Similar triangles ABD and ACE aredrawn outwardly on the sides AB and AC of ABC, so that ABD = ACE and BAD = CAE. CD and BE meet AB and AC at P and Qrespectively. Prove that AP = AQ if and only if

[ABD] · [ACE] = [ABC]2,

where [XY Z] denotes the area of triangle XY Z. (This problem is an ex-tension of Crux 1537.)

Crux 1914.S21.28*. (K.R.S.Sastry)

Let A1A2 · · ·An be a regular n−gon, with M1, M2, . . . , Mn the midpointsof the sides. Let P be a point in the plane of the n−Gon. Prove that

∑PMi ≥ cos

180

n

∑PAi.

Crux 1918.S21.34*. (D.J. Smeenk)

ABC is a triangle with circumcenter O and incenter I, and K, L, N arethe midpoints of BC, CA, AB respectively. Let E and F be the feet of thealtitudes from B and C respectively.

(a) If OK2 = OL2+OM2, show that E, F , O are collinear and determineall possible values of BAC.

(b) If instead OK = OL + OM , show that E, F , I are collinear, anddetermine all possible values of BAC.


Crux 1920.S21.58*. (W.Janous)

Let a, b, c be the sides of a triangle.(a) Prove that for any 0 < λ ≤ 2,

1(1 + λ)2

<(a+ b)(b+ c)(c + a)

(λa+ b+ c)(a+ λb+ c)(a + b+ λc)≤(

22 + λ

)3

,

and that both bounds are best possible.(b) What are the bounds for λ > 2?


D and E are points on sides AB and AC of triangle ABC such thatDE//BC, and P is an interior point of ADE. PB and PC meet DEat F and G respectively. Let O1 and O2 be the circumcenters of PDGand PFE respectively. Prove that AP ⊥ O1O2.


In triangle ABC, cevians AD, BE, CF are equal and concur at point P .Prove that

PA+ PB + PC = 2(PD + PE + PF ).

Crux 1926.S21.67. (W.Pompe)

On sides BC, CA, AB of triangle ABC are chosen points A1, B1, C1 re-spectively, such that triangle A1B1C1 is equilateral. Let o1, o2, o3 and O1,O2, O3 be respectively the incircles and the incenters of triangle AC1B1,BA1C1, CB1A1. If O1C1 = O2C1, show that

(a) B1O3 = B1O1 and A1O2 = A1O3;(b) three external common tangents to the pairs of circles o1, o2, o2, o3,

o3, o1 different from the sides of triangle ABC, have a common point.

Crux 1930.S21.72. (V.Konecny)

T1 is an isosceles triangle with circumcircle K. Let T2 be another isoscelestriangle inscribed in K whose base is one of the equal sides of T1, and whihcoverlaps the interior of T1. Similarly create isosceles triangles T3 from T2,T4 from T3, and so on. Do the triangles Tn approach an equilateral triangleas n→ ∞?



M is the midpoint of side BC of ABC,and Γ is the circle with diameterAM . D and E are the other intersections of Γ with AB andAC respectively.Let P be the point such that PD and PE are tangent to Γ. Prove thatPB = PC.



Two externally tangent circles of radii R1 and R2 are internally tangent toa semicircle of radius 1, as in the figure. Prove that

R1 +R2 ≤ 2(√

2 − 1).


Given an ellipse which is not a circle, prove or disprove that the locus of themidpoints of sufficiently small constant length chords is another ellipse.

Crux 1937.S21.102*. (D.J. Smeenk)

Triangle ABC has circumcenter O, orthocenter H, and altitudes AD, BE,and CF (with D on BC etc). Suppose OH//AC.

(a) Show that EF , FD, and DE are in arithmetic progression.(b) Determine the possible values of angle B.

Crux 1939.S21.105. (C.J.Bradley)

Let ABC be an acute-angled triangle with circumcenter O, incenter I, andorthocenter H. Let AI, BI, CI meet BC, CA, AB at U , V , W , and AH,BH, CH meet BC, CA, AB respectively in D, E, F . Prove that O is aninterior point of triangle UVW if and only if I is an interior point of triangleDEF .



ABCD is a convex quadrilateral, and O is the intersection of its diagonals.Suppose that the area of the (nonconvex) pentagon ABOCD is equal to thearea of triangle OBC. Let P and Q be the points on BC such that OP//ABand OQ//DC. Prove that

[OAB] + [OCD] = 2[OPQ],

where [XY Z] denotes the area of triangle XY Z.


In triangle ABC, the median AD is the geometric mean of AB and AC.Prove that

1 + cosA =√

2| cosB − cosC|.

Crux 1947.S21.139. (D.J. Smeenk)

Triangle ABC has incenter I and centeroid G. The line IG intersects BC,CA, AB in K, L, M respectively. The line through K parallel to CAintersects the internal bisector of angle BAC at P . The line through Lparallel to AB intersects the internal bisector of angle CBA in Q. The linethrough M parallel to BC intersects the internal bisector of angle ACB inR. Show that BP , CQ, AR are parallel.

Crux 1949.S21.141. (F.Ardila)

Let D, E, F be points on the sides BC, CA, AB respectively of triangleABC, and let R be the circumradius of ABC. Prove that

(1AD

+1BE

+1CF

)(DE + EF + FD) ≥ AB +BC + CA

R.


ABCD is a cyclic quadrilateral, and P is the intersection of the diagonalsAC and BD. A line through P meets AB and CD at E and F respectively.Let O1 and O2 be the circumcenters of PAB and PCD, and let Q be thepoint on O1O2 such that PQ ⊥ . Prove that EP : PF = OQ : QO2.



The convex cyclic quadrilateral ABCD is such that each of its diagonalsbisects one angle and trisects the opposite angle. Determine the angles ofABCD.

Crux 1954.S21.166*. (V.N.Murty)

Let ABC be a triangle with A < π2 and B ≤ C. The tangents to the

circumcircle of triangle ABC at B and C meet at D. Put θ = OAD, whereO is the circumcenter. Prove that

2 tan θ = cotB − cotC.



In a semicircle of radius 4, there are three tangent circles as in the figure.Prove that the radius of the smallest circle is at most

√2 − 1.

Crux 1960.S21.174*. (W.Pompe)

Two perpendicular linesand a circle pass through a common point. Threeline segments AB, CD, EF , with endpoints on the two perpendicular lines,are drawn tangent to C at their midpoints. Prove that the length of onesegment is equal to the sum of the lengths of the other two.


ABC is an isosceles triangle with AB = AC. We denote the circumcircle ofABC by Γ. Let D be the point such that DAA and DC are tangent to Γat A and C repsectively. Prove that DBC ≤ 30.


In triangle ABC, one pair of trisectors of the angles B and C meet at theorthocenter. Show that the other pair of trisectors of these angles meet atthe circumcenter.

Crux 1965.S21.207*. (Ji Chen)

Let P be a point in the interior of triangle ABC, and let the lines AP , BP ,CP intersect the opposite sides at D, E, F respectively.


(a) Prove or disprove that

PD · PE · PF ≤ R3

8,

where R is the circumradius of triangle ABC. Equality holds when ABC isequilateral and P is its center.

(b) Prove or disprove that

PE · PF + PF · PD + PD · PE ≤ 14

max(a2, b2, c2),

where a, b,c are the sides of the triangle. Equality holds when ABC isequilateral and P is its center, and also when P is the midpoint o thelongest side of triangle ABC.

Crux 1967.S21.209*. (C.J.Bradley)

ABC is a triangle and P is a point in its plane. The lines through P parallelto the medians of the triangle meet the opposite sides in points U , V , W .Describe the set of points for which U , V , W are collinear.

Crux 1971.S21.240*. (T.Seimiya)

A convex quadrilateral ABCD with AC = BD is inscribed in a circle withcenter O, and E is the intersection of the diagonals AC and BD. Let P bean interior point of ABCD such that

PAB + PCB = PBC + PDC = 90.

Prove that O, P , E are collinear.

Crux 1973.S21.245. (K.R.S.Sastry)

Triangle ABC is inscribed in a circle. The chord AD bisects BAC. Assumethat AB =

√2BC =

√2AD. Determine the angles of triangle ABC.

Crux 1977.S21.250*. (D.J. Smeenk)

Triangle ABC has circumcenter O. Let be the line through O parallel toBC, and let P be a variable point on . RThe projections of P on BC, CA,AB are Q, R, S respectively. Show that the circle passing through Q, R,and S passes through a fixed point, independent of P .


Crux 1980.S21.279*. (I.Beck and N.Nejlegaard)

Find all sets of four points in the plane so that the sum of the distancesfrom each of the points to the other three is a constant.

Crux 1981.S21.255* (T.Seimiya)

ABC is an obtuse triangle with A > 90. Let I and O be the incenter andthe circumcenter of ABC. Suppose that [IBC] = [OBC], where [XY Z]denotes the area of triangle XY Z. Prove that

[IAB] + [IOC] = [ICA] + [IBO].


A convex quadrilateral ABCD has an inscribed circle with center I and alsohas a circumcircle. Let the line parallel to AB through I meet AD in A′

and BC in B′. Prove that the length of A′B′ is a quarter of the perimeterof ABCD.


Crux 1985

See Crux 2355.

Crux 1987.S21.283*;22(6)276. (H.G—’ulicher)

In the figure, B2C1//A1A2, B3C2//A2A3, and B1C3//A3A1. Prove thatB2C1, B3C2 and B1C2 are concurrent if and only if

A1C3

C3B3· A2C1

C1B1· A3C2

C2B2= 1.


Ω is a fixed circle with center O. Let M be the foot of the perpendicular fromO to a fixed line , and let P be a variable point on Ω. Let Γ be the circlewith diameter PM , intersecting Ω and again at X and Y respectively.Prove that the line XY always passes through a fixed point.

Crux 1993.S21.308*. (W.Pompe)

ABCD is a convex quadrilateral inscribed in a circle Γ. Assume that A, Band Γ are fixed and C, D are variable, so that the length of the segmentCD is constant. X, Y are the points on the rays AC and BC respectively,such that AX = AD and BY = BD. Prove that the distance between Xand Y remains constant.

See also Crux 1902.


Crux 1997.S21(9)317*. (C.J.Bradley)

ABC is a triangle which is not equilateral, with circumcenter O and ortho-center H. Point K lies on OH so that O is the midpoint of HK. AK meetsBC in X, and Y , Z are the feet of the perpendiculars from X onto the sidesAC, AB respectively. Prove that AX, BY , CA are concurrent or parallel.

Crux 1999.S21(9)320*. (R.Mqrquez)

Let ABC be a variable isosceles triangle with constant side a = b andvariable side c. Denote the median, angle bisector, and altitude, measuredfrom A to the opposite side, by m, w, and h respectively. Find

lim c→ am− h

w − h.

Crux 2001.S21.345*. (T.Seimiya)

Three similar triangle DBC, ECA, FAB are drawn outwardly on the sidesof triangle ABC, such that DBC = ECA = FAB and DCB = EAC = FBA. Let P be the intersection of BE and CF , Q that ofAD and BE, and R that of AD and BE. Prove that

QR

AD=RP

BE=PQ

CF.


Crux 2005.S21.350*. (Klamkin)

(a) Let a, b, c be the vectors from the cirucmcenter of a triangle ABC tothe respective vertices. Prove that

(b+ c)|b− c||b+ c| +

(c+ a)|c− a||c+ a| +

(a+b)|a−b||a+b| = 0. (2)

(b) Suppose that a, b, c are vectors from a point P to the respectivevertices of a triangle such that (1) holds. Must P be the circumcenter ofthe ABC?

Crux 2008.S22.40*. (J.H.Huang)

Let I be the incenter of triangle ABC, and suppose there is a circle withcenter I which is tangent to each of the excircles of ABC. Prove thatABC is equilateral.

Solution. (partial) If this circle touches each of the excircles externally,then it must be the nine-point circle. Now, since the nine-point coincideswith the incenter, the triangle must be equilateral.

Crux 2010.S22.40*. (E.Kuczma)

In triangle ABC with C = 2 A, line CD is the internal angle bisector withD on AB, Let S be the center of the circle tangent to line CA (producedbeyond A) and externally tangent to the circumcircles of triangles ACD andBCD. Prove that CS ⊥ AB.


ABC is a triangle with incenter I. BI and CI meet AC and AB at D andE respectively. P is the foot of the perpendicualr from I to DE, and IPmeets BC at Q. Suppose that IQ = 2IP . Find angle A.


The number of primitive Pythagorean triangle with a fixed inradius is alwaysa power of 2. Proof. A primitive Pythagorean triangle (m2 −n2, 2mn,m2 +n2) has inradius r = n(m − n). If a given integer r has exactly k distinctprime divisors, then there are 2k ways of factoring r in the form n(m−n) withn and m − n relatively prime. Each of these factorizations gives relativelyprime integers m > n leading a primitive Pythagorean triangle.


The only r as the inradius of a unique primitive Pythagorean triangle isr = 1, and the triangle has sides 3,4,5.

Crux 2013.S22.44. (W.Pompe)

Given a convex n−gon A1A2 · · ·An (n ≥ 3) and a point P in its plane.Assume that the feet of the perpendiculars from P to the lines A1A2, A2A3

, . . . , AnA1 all lie on a circle with center O.(a) Prove that if P belongs to the interior of the n−gon, then so does O.(b) Is the converse to (a) true ?(c) Is (a) still valid for nonconvex n−gons ?

Crux 2015.S22(1)47,125*;24(5)305*. (S.C. Shi and Ji Chen)

Prove that

(sinA+ sinB + sinC)(1A

+1B

+1C

) ≥ 27√

3π

,

where A, B, C are the angles of a triangle measured in radians.

Crux 2017.S22.82. (D.J. Smeenk)

We are given a fixed circle κ and two fixed points A and B not lying on κ.A variable circle through A and B intersects κ in C and D.Show that theratio

AC ·ADBC ·BD

is constant.

Crux 2019.S22.85*.(P.Penning)

In a plane are given a circle C with a diameter l and a point P within C butnot on l. Construct the equilateral triangles that have one vertex at P , oneon C, and one on l.


P is a variable interior point of triangle ABC, and AP , BP , CP meet BC,CA, AB at D, E, F respectively. Find the locus of P so that

[PAF ] + [PBD] + [PCE] =12[ABC],




It is a known result that if P is any point on the circumcircle of a giventriangle ABC with orthocenter H, then PA2 + PB2 + PC2 − PH2 is aconstant. Generalize this result to an n−dimensional simplex.

Crux 2027. (D.J. Smeenk)

Quadrilateral ABCD is inscribed in a circle Γ, and has an incircle as well.EF is a diameter of Γ with EF ⊥ BD. BD intersects EF in M and AC inS. Show that AS : SC = EM : MF .

Crux 2029.S22.129*. (J.H.Chen)

ABC is a triangle with area F and internal angle bisectors wa, wb, wc. Proveor disprove that

wbwc + wcwa + wawb ≥ 3√

3F.

Crux 2031.S22(3)135. (T.Seimiya)

Suppose that α, β, γ are acute angles such that

sin(α− β)sin(α+ β)

+sin(β − γ)sin(β + γ)

+sin(γ − α)sin(γ + α)

= 0.

Prove that at least two of α, β, γ are equal.

Crux 2033.S22(3)137*. (K.R.S.Sastry)

The sides AB, BC, CD., DA of a convex quadrilateral ABCD are extendedin that order to th points P , Q, R, S such that BP = CQ = DR = AS. IfPQRS is a square, prove that ABCD is also a square.

Crux 2035.S22(4)172. (V.Konecny)

If the locus of a point E in an ellipse with fixed foci F and G, prove thatthe locus of the incenter of triangle EFG is another ellipse.

Crux 2039.S22(3)144. (D.Zhou)


sinAB

+sinBC

+sinCA

≥ 9√

32π

.



P is an interior point of triangle ABC. AP , BP , CP meet BC, CA, ABat D, E, F respectively. Let M and N be points on segments BF and CErespectively so that BM : MF = EN : NC. Let MN meet BE and CF atX and Y respectively. Prove that MX : Y N = BD : DC.

Crux 2043.S22(4)176*. (A.A.Yagubyants)

What is the locus of a point interior to a fixed triangle that moves so thatthe sum of its distances to the sides of the triangle remains constant?

Crux 2047.S22(4)181*. (D.J. Smeenk)

ABC is a nn-equilateral triangle with circumcentr O and incenter I. D isthe foot of the altitude from A to BC. Suppose that the circumradius Requals the radius ra of the excircle to BC. Show that O, I, D are collinear.


A convex quadrilateral ABCD is inscribed in a circle Γ with center O. Pis an interior point of ABCD. Let O1, O2, O3, O4 be the circumcenters oftriangles PAB, PBC, PCD, PDA respectively. Prove that the midpointsof O1O3, O2O4 and OP are collinear.

Crux 2053. S22(4)187. (J.Kotani)

A figure consisting of two equal and externally tangent circles is inscribedin an ellipse. Find the eccentricity of the ellipse of minimum area.

Crux 2055.S22(4)189. (H.Gulicher)

In triangle ABC lete D be the point on the ray BC and E on CA such thatBD = CE = AB, let be the line through D parallel to AB. IF M is theintersection of and BE, and F that of CM and AB, prove that

BA3 = AE · BF · CD.

Crux 2057.S22(4)190. (J.Ciach)

Let P be a point inside an equilateral triangle ABC, and let Ra, Rb, Rc andra, rb, rc denote the distances of P from the vertices and edges respectively


of the triangle. Prove or disprove that

(1 +raRa

)(1 +rbRb

)(1 +rcRc

) ≥ 278.

Equality holds if P is the center of the triangle.See also Crux 2073.

Crux 2061.S22(5)230 (T.Seimiya)

ABC is a triangle with centroid G, and P is a variable interior point ofABC. Let D, E, F be points on sides BC, CA, AB respectively such thatPD//AG, PE//BG and PF//CG. Frove that [PAF ] + [PBD] + [PCE] isconstant, where [XY Z] denotes the area of triangle XY Z.


Triangle ABC has a right angle at C.(a) Prove that the three ellipses having foci at two vertices of the given

triangle, while passing through the third, all share a common point.(b) Prove that the principal vertices of the ellipses of part (a), that is

the points where an ellipse meets the axis through its foci) form two pairsof collinear triples.

See also Crux 728.S9.116*.

Crux 2067.S22(6)277*. (M.Stupel and V.Oxman)

Triangle ABC is inscribed in a circle Γ. Let AA1, BB1, CC1 be the bisectorsof angles A, B, C, with A1, B1, C1 on Γ. Prove that the perimeter of thetriangle is equal to

AA1 cosA

2+BB1 cos

B

2+ CC1 cos

C

2.

Crux 2069.S22(6)278*. (D.J. Smeenk)

M is a variable point of side BC of triangle ABC. A line through Mintersects the line AB in K and AC in L so that M is the midpoint of thesegment KL. Point K ′ is such that ALKK ′ is a parallelogram. Determinethe locus of K ′ as M moves on segment BC.



P is an interior point of an equilateral triangle ABC so that PB = PC,and BP and CP meet AC and AB at D and E respectively. Suppose thatPB : PC = AD : AE. Find angle BPC.

Crux 2073.S22(6)282. (J.Ciach)

Let P be an interior point of an equilateral triangle A1A2A3, and let R1 =PA1, R2 = PA2, R3 = PA3. Prove or disprove that

R1R2R3 ≤ 98R3.

Equality holds if P is the midpoint of a side.See also Crux 1895 and 2057.


ABC is a triangle with A < B < C, and I its incenter. BCL, ACM andABN are the sides of the triangle with L on BC produced etc., and theponts L, M , N chosen so that

CLI =12(C −B), AMI =

12(C −A), BNI =

12(B −A).

Prove that L, M , N are collinear.

Crux 2079.S22(7)325. (C.Sanchez-Rubio and I.B.Penyagolosa)

An ellipse is inscribed in a rectangle. Prove that the contact points of theellipse with the sides of the reectangle lie on the rectangular hyperbola whichpasses through the foci of the ellipse and whose asymptotes are parallel tothe sides of the rectangle.


ABC is a triangle with A > 90, and AD, BE and CF are its altitudes(with D on BC, etc.). Let E′ and F ′ be the feet of the perpendiculars fromE and F to BC. Suppose that 2E′F ′ = 2AD +BC. Find A.


Crux 2084.S22(7)330. (M.S.Klamkin)

Prove that ∑cos

B

2cos

C

2≥ 1 − 2 cos

A

2cos

B

2cos

C

2,

where A, B, C are the angles of a triangle.

Crux 2089.S22(8)366. (M.A.Covas)

Let ABCD be a trapezoid with ABCD and let X be a point on segment AB. Put P = CB∩AD, Y = CD∩PX,R = AY ∩BD, and T = PR ∩AB. Prove that

1AT

=1AX

+1AB

.

Crux 2091.S22(8)367*. (T.Seimiya)

Four points A, B, C, D are on a line in this order. We put AB = a, BC = b,CD = c. Equilateral triangles ABP , BCQ and CDR are constructed onthe same side of the line. Suppose that PQR = 120. Find the relationbetween a, b and c.

Crux 2093.S22(8)371. (W.Janous)

LetA, B, C be the angles (in radians) of a triangle. Prove or disprove

(sinA+ sinB + sinC)(1

π −A+

1π −B

+1

π − C) ≤ 27

√3

4π.

Crux 2096.S22(8)374. (D.J. Smeenk)

Triangle A1A2A3 has circumcircle Γ. The tangents at A1, A2, A3 to Γintersect (the extensions of) A2A3, A3A2, A1A2 respectively in B1, B2,B3. The second tangent to Γ through B1, B2, B3 touch Γ at C1, C2, C3

respectively. Show that A1C1, A2C2, A3C3 are concurrent.


Crux 2101.S23(1)49. (J.Chen)

Prove that for any k ≤ 1,

∑ ak

A≥ 3π

∑ak,

where the sums are cyclic. [The case k = 1 is known; see item 4.11 (p.170)of Mitrinovic et. al].


ABC is a triangle with incenter I. Let P and Q be the feet of the perpen-diculars from A to BI and CI respectively. Prove that

AP

BI+AQ

CI= cot

A

2.

Crux 2103.S23(1).52. (T.Seimiya)

ABC is a triangle. Let D be the point on side BC produced beyond B suchthat BD = BA, and let M be the midpoint of AC. The bisector of ABCmeets DM at P . Prove that BAP = ACB.

Crux 2106.S23(1).55. (YANG Kechang)

A quadrilateral has sides a, b, c, d (in that order) and area F . Prove that

2a2 + 5b2 + 8c2 − d2 ≥ 4F.


Crux 2107.S23(1)57*. (D.J. Smeenk)

Triangle ABC is not isosceles nor equilateral, and has sides a, b, c. D1 andE1 are points of BA and CA or their productions so that BD1 = CE1 = a.D2 and E2 aere points of CB and AB or their productions so hat CD2 =AE2 = b. Show that D1E1//D2E2.


Crux 2109.S23(1)60*. (V.Oxman)

In the plane are given a triangle and a circle passing through two of thevertices of the triangle and also through the incenter of the triangle. (Theincenter and the center of the circle are not given). Construct, using onlyan unmarked ruler, the incenter.

Remark. The “easy” construction fails if the triangle is isosceles, withthe circle passing through the base vertices. In that case, the incenter is themidpoint of the arc, and cannot be constructed with an unmarked ruler.


ABCD is a square with incircle Γ. A tangent to Γ meets the sides ABand AD and the diagonal AC at P , Q, and R respectively. Prove that

AP

PB+AR

RC+AQ

QD= 1.

Crux 2116.S23(2)116. (YANG Kechang)

A triangle has sides a, b, c and area F . Prove that

a3b4c5 ≥ 25√

5(2F )6

27.



ABC is a triangle with AB > AC, and the bisector of A meets BC at D.Let P be an interior point of the side AC. Prove that BPD < DPC.

Crux 2120.S23(2)122. (M.E.Kuczma)

Let A1A3A5 and A2A4A6 be nondegenerate triangles in the plane. Fori = 1, . . . , 6, let i be the perpendicular from Ai to the line Ai−1Ai+1 (where,of course, A0 = A6 and A7 = A1). If 1, 3, 5 concur, prove that 2, 4, 6also concur.

Crux 2124.S23(3)171*. (C.Shevlin)

Suppose that ABCD is a quadrilateral with CDB = CBD = 50 and CAB = ABD = BCD. Prove that AD ⊥ BC.



ABC is an acute triangle with circumcenter O, and D is a point on theminor arc AC of the circumcircle D = A,C). Let P be a point on the sideAB such that ADP = OBC, and let Q be a point onth side BC suchthat CDQ = OBA. Prove that DPQ = DOC and DQP = DOA.

Crux 2128.S23(3)178. (T.seimiya)

ABCD is a square. Let P and Q be interior points onthe sides BC and CDrespectively, and let E and F be the intersetions of PQ with AB and ADrespectively. Prove that

π ≤ PAQ+ ECF <54π.

Crux 2130.S23(3)179*. (D.J. Smeenk)

A and B are fixed points, and is a fixed line passing through A. C is avariable point on staying on one side of A. The incircle of ABC touchesBC at D an AC at E. Show that the line DE passes through a fixed point.

Solution. Set up an oblique coordinate system with A as origin and thelines l and AB as axes. Suppose the segments AB, AE, and EC have lengthsc, x, and y respectively. Then the points B, E, and C have coordinates (0, c),(x, 0), and (x+ y, 0) respectively. If the incircle touches AB at the point F ,then it is easy to see that AF = AE = x, so that

BD = BF = c− x.

Note also that CD = CE = y. It follows that the point D has coordinates

1c− x+ y

[y(0, c) + (c− x)(x+ y, 0)] =1

c− x+ y

((c− x)(x+ y), cy

).

A typical point on the line DE has coordinates tD + (1 − t)E for somet(= −1). Explicitly, this is the point

(x+

ty

c− x+ y· (c− 2x),

ty

c− x+ y· c).

By choosing t such that tyc−x+y = 1

2 , we obtain the point 12(c, c) independent

of x and y. This fixed point can be identified as the midpoint of the segmentBB′, where B′ = (c, 0) is the point on l such that AB′ = AB = c.


Having located the fixed point by coordinate geometry, we can now givea simple synthetic proof.

Alternative Solution Let B′ be the point on l such that AB′ = AB = c.Suppose AE = x and EC = y (as above). Regarding DE as a transversalof the triangle BB′C intersecting BB′ at the point P , we have CD = y,DB = c−x, B′E = AB′−AE = c−x, and EC = y. By Menelau’s theorem,

BP

PB′ ·B′EEC

· CDDB

= −1.

Here,

(i) B′EEC = − c−x

y , E dividing B′C externally, and

(ii) CDDB = y

c−x .

It follows that BPPB′ = 1, and DE passes through the midpoint of the

segment BB′.

Crux 2133.S23(4)249. (K.R.S.Sastry)

Similar non-square rectangels are placed outwardly on the sides of a parallel-ogram π. Prove that the centers of these rectangles also form a non-squarerectangle if and only if π is a non-square rhombus.

Crux 2136.S23(3)185. (G.P.Henderson)

Let a, b, c be the lengths of the sides of a triangle. Given the values ofp =

∑a and q =

∑ab, prove that r = abc can be estimated with an error

of at most 126r.

Crux 2137.(corrected 22(7))S23(3)187*.. (A.A.Yagubyants)

Three circles of (equal) radius t passes through a point T , andareeach insidetriagle ABC and tangent t two of its sides. Prove that

(i) t = 2RR+2 ;

(ii) T lies on the line segment joining the centers of the circumcircle andthe incircle of ABC.

See also Crux 694.



ABC is an acute angle triangle with circumcenter O. AO meets the circleBOC again at A′, BO meets the circle COA again at B′, and CO meetsthe circle AOB again at C ′. Prove that [A′B′C ′] ≥ 4[ABC], where [XY Z]denotes the area of triangle XY Z.

Crux 2139.(corrected 22(5))S23(3)190. (W.Pompe)

Point P lies inside triangle ABC. Let D, E, F be the orthogonal projectionsfrom P onto the lines BC, CA, AB respectively. Let O′, and R′ denote thecircumcenter and circumradius of the triangle DEF respectively. Prove that

[ABC] ≥ 3√

3R′√R′2 −O′P 2,



A1A2A3A4 is a quadrilateral. Let B1, B2, B3, B4 be points on the sidesA1A2, A2A3, A3A4, A4A1 respectively, such that

A1B1 : B1A2 = A4B3 : B3A3

andA2B2 : B2A3 = A1B4 : B4A4.

Let P1, P2, P3 and P4 be points on B4B1, B1B2, B2B3, and B3B4 such that

P1P2//A1A2, P2P3//A2A3, P3P4//A3A4.

Prove that P4P1//A4A1.

Crux 2142.S23(4)252. (V.Oxman)

In the plane are given an arbitrary quadrangle and bisectors of three of itsangles. Construct, using only an unmarked ruler, the bisector of the fourthangle.


ABC is a triangle with AB > AC, and the bisector of A meets BC atD. Let P be an interior point on the segment AD, and let Q and R be thepoints of intersection of BP and CP with sides AC and AB respectively.Prove that PB − PC > RB −QC > 0.



Suppose that AD, BE and CF are the altitudes of triangle ABC. Suppoethat L, M , N are points on BC, CA, AB respectively, such that BL = DC,CM = EA, AF = NB. Prove that

1. the perpendiculars to BC, CA, AB at L, M , N respectively areconcurrent;

2. the point of concurrency lies on the Euler line of triangle ABC.

Crux 2149.S23(5)306*. (J.B.Romero Marquez)

Let ABCD be a convex quadrilateral and O the point of intersection of thediagonals AC and BD. Let A′B′C ′D′ be the quadrilateral whose verticesA′, B′, C ′, D′ are the feet of the perpendiculars drawn from the point O tothe sides BC, CD, DA, AB respectively.

Prove that ABCD is an inscribed (cyclic) quadrilateral if and only ifA′B′C ′D′ is a circumscribing quadrilateral (A′B′, B′C ′, C ′D′, D′A′ aretangents to a circle).

Crux 2151.S23(5)310*. (T.Seimiya)

ABC is a triangle with B = 2 C. Let H be the foot of the perpendicularfrom A to BC, and let D be the point on the side BC where the excircletouchese BC. Prove that AC = 2HD.

Crux 2154.S23(5)314*. (K.R.S.Sastry)

In a convexpentagon, the medians are concurrent. If the concurrence pointsections each median in the same ratio, find its numerical value. (A medianof a pentagon is the line segment between a vertex and the midpoint of thethird side from the vertex).

Crux 2156.S23(5)318*. (H.T.Wee)

ABCD is a convex quadrilateral with perpedicular diagonals AC and BD.S and Y are points in the interior of sides BC and AD resepectively suchthat

BX

CX=BD

AC=DY

AY.

Evaluate BC·XYBX·AC .



ABC is a triangle with A < 90. Let P be an interior point of ABCsuch that BAP = ACP and CAP = ABP . Let M and N be theincenters of ABP and ACP respectively, and let R1 be the circumradiusof AMN . Prove that

1R1

=1AB

+1AC

+1AP

.

Crux 2162.S23(6)373*. (D.J. Smeenk)

In ABC, the Cevian lines AD, BE, CF concur at P . [XY Z] is the areaof XY Z. PRove that

[DEF ]2[ABC]

=PD

PA· PEPB

· PFPC

.

Crux 2164.S23(6)377*. (T.Seimiya)

Let D be a point on the side BC of triagle ABC, and let E, F be theincenters of triangle ABD and ACD respectively. Suppose that B, C, E,F are concyclic. Prove that

AD +BD

AD + CD=AB

AC.

Crux 2165.S23(6)378. (H.T.Wee)

Given a triangle ABC, prove that there exists a unique pair of points P andQ such that the triangles ABC, PQC and PBQ are directly similar; thatis, ABC = PQC = PBQ and BAC = QPC = BPQ, and the threesimilar triangle have the same orientation. Find a euclidean constructionfor the points P and Q.

Crux 2166.S23(6)380*. (K.R.S.Sastry)

In a right - angled triangle, establish the existence of a unique interior pointwith the property that the line through the point perpendicular to any sidecuts off a triangle of the same area.


Crux 2169.S23(7)434. (D.J. Smeenk)

AB is a fixed diameter of circle C1 := OR. P is an arbitrary point on itscircumference. Q is the projection onto AB of P . Circle P1PQ intersects C1

at C and D. CD intersects PQ at E. F is the midpoint of AQ. FG ⊥ CD,where G ∈ CD. Show that

(1) EP = EQ = EG;(2) A, G and P are collinear.

Crux 2171.S23(7)436*. (J.B. Romero Marquez)

Let P be an arbitrary point taken on an ellipse with foci F1 and F2, anddirectrices d1 and d2 respectively. Draw the straight line through P whichis parallel to the major axis of the ellipse. This lien intersects d1 and d2

at points M and N respectively. Let P ′ be the point where MF1 intersectsNF2. Prove that the quadrilateral PF1P

′F2 is cyclci. Does the result alsohold in the case of a hyperbola?


ABCD is a convex quadrilateral, with P the intersection of its diagonalsand M the midpoint of AD. MP meets BC at E. Suppose that BE :EC = AB2 : CD2. Characterize quadrilateral ABCD.


If A, B, C are the angles of a triangle, prove that

sinA sinB sinC ≤ 8(sin3A cosB cosC + sin3B cosC cosA+ sin3 C cosA cosB

≤ 3√

3(cos2A+ cos2B + cos2C).


Suppose A, B, C are the angles of a triangle, and that k, , m ≥ 1. Showthat

0 < sink A sinB sinmC≤ kkmmSS/2(Sk2 + P )−k/2(S2 + P )−/2(Sm2 + P )−m/2,

where S = k + +m and P = km.


Crux 2186.S23(8)519*. (V.N.Murty)

GI2 =1

9(a+ b+ c)

∑(a− b)(b− c)(b+ c− a).

Deduce from thisGI2 =

19(s2 + 5r2 − 16Rr).

Crux 2188.S23(8)522. (V.Oxman)

Suppose that a, b, c are the sides of a triangle with semiperimeter s andarea . Prove that

1a

+1b

+1c<

s

.

Solution. Note that a = (s− b) + (s − c), and

1a

=1

(s− b) + (s− c)≤ 1

2√

(s− b)(s − c)=√s(s− a)2 <

2s− a

4 .

Here, equality does not hold since s = s− a. Similarly,

1b<

2s− b

4 and1c<

2s− c

4 .

It follows that

1a

+1b

+1c<

(2s − a) + (2s − b) + (2s − c)4 =

4s4 =

s

.


The incircle of a triangle ABC touches BC at D. Let P and Q be variablepoints on sides AB and AC respectively such that PQ is tangent to theincircle. Prove that the area of triangle DPQ is a constant multiple ofBP · CQ.

Crux 2190.S23(8)525*. (W.Janous)

Determine the range of

sin2A

A+

sin2B

B+

sin2C

C

where A, B, C are the angles of a triangle.



Prove or disprove that it is possible to find a triangle ABC and a transversalNML with N lying between A and B, M lying between A and C, and Llying on BC produced, such that BC, CA, AB, NB, MC, NM , ML, andCL are all of integer length, andNMCB is a cyclic inscriptible quadrilateral.

Crux 2198.S24(1)49*. (V.N.Murty)

Prove that if a, b, c are the lengths of the sides of a triangle,

(b− c)2(2bc

− 1a2

) + (c− a)2(2ca

− 1b2

) + (a− b)2(2ab

− 1c2

) ≥ 0,

with equality if and only if a = b = c.


ABCD is a convex quadrilateral, and O is the intersection of its diagonals.Let L, M , N be the midpoints of DB, BC, CA respectively. Suppose thatAL, OM , DN are concurrent. Show that either AD//BC or [ABCD] =2[OBC], where [F ] denotes the area of figure F .

Crux 2202.S24(1)54*. (W.Janous)

Supoose that n ≥ 3. Let A1 · · ·An be a convex n−gon. Determine thegreatest constant Cn such that

n∑k=1

1Ak

≥ Cn

n∑k=1

1π −Ak

.

Determine when equality occurs.

Crux 2203.S24(1)56. (W.Janous)

Let ABCD be a quadrilateral with incircle I. Dentoe by P , Q, R and Sthe points of tangency of sides AB, BC, CD and DA, respectively with I.Determine all possible values of (PR,QS) such that ABCD is cyclic.

Crux 2204.S24(1)57. (S.Arslanaglf)

For triangle ABC such that R(a+ b) = c√ab, prove that

r <310a.


Crux 2205.S24(1)58*. (V.Konecny)

Find the least positive integer n such that the expression

sinn+2A sinn+1B sinnC

has a maximum which is a rational number. Here, A, B, C are the anglesof a variable triangle.

Crux 2209.S24(2)112*. (M.A.Caberzon Ochoa)

Let ABCD be a cyclic quadrilateral having perpendicular diagonals crossingat P . Project P onto the sides of the quadrilateral.

1. Prove that the quadrilateral obtained by joining these four projectionsis inscribable and circumscribable.

2. Prove that the circle which passes through these four projections alsopasses through the midpoints of the sides of the given quadrilateral.

Crux 2215.S24(2)121. (T.Chronis)

Let P be a point inside a triangle ABC. Determine P such that PA+PB+PC is a maximum.

See also CMJ346.872.S892. There it is shown that the maximum oc-curs at a vertex where two longest sides of the triangle meet. There is nomaximum inside the triangle.

Crux 2224.S24(3)184. (W.Pompe)

Point P lies inside triangle ABC. Triangle BCD is erected outwardly onside BC such that BCD = ACP and CBD = ABC. Prove that ifthe area of quadrilateral PBDC is equal to the area of triangle ABC, thentriangle ACP and BCD are similar.

Crux 2230.S24(3)191. (W.Pompe)

Triangels BCD and ACE are constructed outwardly on sides BC and CAof triangle ABC such that AE = BD and BDC + AEC = 180. Thepoint F is chosen to lie on the segment AB so that

AF

FB=DC

CE.

Prove thatDE

CD + CE=EF

BC=FD

AC.


Crux 2231.S24(4)242. (H.Gulicher)

In quadrilateral P1P2P3P4, suppose that the diagonals intersect at the pointM = Pi, (i = 1, 2, 3, 4). Let MP1P4 = α1, MP3P4 = α2, MP1P2 = β1,and MP3P2 = β2. Prove that

λ13 :=|P1M ||MP3| =

cotα1 ± cot β1

cotα2 ± cot β2,

where the +(−) sign holds if the line segment PqP3 is located inside (outside)the quadrilateral.

Crux 2234(=2287).S24(4)247;(8)525. (V.Oxman)

Given triangle ABC, its centroid G and its incenter I, construct, using onlyan unmarked ruler, its orthocenter H.

Crux 2235.S24(4)249. (D.J. Smeenk)

Triangle ABC has angle CAB = 90. Let Γ1(O,R) be the cirucmcircle andΓ2(T, r) be the incircle. The tangent to Γ1 at A and the polar line of Awith respect to Γ2 intersect at S. The distance from S to AC and AB aredenoted by d1 and d2 respectively. Show that

(a) ST//BC,(b) |d1 − d2| = r.

Crux 2236.S24(4)250*. (V.Oxman)

Let ABC be an arbitrary triangle and let P be an arbitrary point in theinterior of the circumcircle of triangle ABC, Let K, L, M denote the feetof the perpendiculars from P to the lines AB, BC, CA respectively. Provethat

[KLM ] ≤ 14[ABC].

Crux 2237.S24(4)251*. (M.D.Visiliou)

ABCD is a square with incircle Γ. Let be a tangent to Γ. Let A′, B′, C ′,D′ be points on such that AA′, BB′m CC ′, DD′ are all perpendicular to. Prove that AA′ · CC ′ = BB′ ·DD′.


Crux 2240.S24(5)312. (V.Oxman)

Let ABC be an arbitrary triangle with the points D, E, F on the sides BC,CA, AB respectively. so that BD

DC ≤ BFFA ≤ 1 and AE

EC ≤ AFFB . Prove that

[DEF ] ≤ 14[ABC]

with equality if and only if two of the three points D, E, F (at least) aremidpoints of the corresponding sides.

Crux 2241.S24(5)313*. (T.Seimiya)

Triangle ABC (AB = AC) has incenter I and circumcenter O. The incircletouches BC at D. Suppose IO ⊥ AD. Prove that AD is a symmedian oftriangle ABC. (A symmedian is the reflection of the median in the internalangle bisector).

Crux 2242.S24(5)314*. (K.R.S.Sastry)

ABCD is a parallelogram. A point P lies in the plane such that(1) the line through P parallel to DA meets DC at K and AB at L,(2) the line through P parallel to AB meets AD at M and BC at N ,

and(3) the angle between KM and LN is equal to the non-obtuse angle of

the parallelogram.Find the locus of P .


ABC is a triangle and D is a point on AB produced beyond B such thatBD = AC, and E is a point on AC produced beyond C such that CE = AB.The perpendicular bisector of BC meets DE at P . Prove that BPC = BAC.

Crux 2246.S24(5)318*. (D.J. Smeenk)

Suppose that G, I and O are the centroid, the incenter, and the circumcenterof a non-equilateral triangle ABC. The line through B, perpendicular toOI intersects the bisector of angle BAC at P . The line through P , parallelto AC intersects BC at M . Show htat I, G, M are collinear.



ABC is a scalene triangle with incenter I. Let D, E, F be the points whereBC, CA, AB are tangent to the incircle respectively, and let L, M , N be themidpoints of BC, CA, AB respectively. Let , m, n be the lines through D,E, F parallel to IL, IM , IN respectively. Prove that , m, n are concurrent.

Crux 2251(=2288).S24(6)373;(8)525. (V.Oxman)

In the plane you are given a circle (but not its center), and points A, K, B,D, C on it, so that arc AK = arc KB and arc BD = arc DC. Construct,using oly an unmarked straightedge, the mid-point of arc AC.


Prove that the nine-point circle of a triangle trisects a median if and onlyif the side length of the triangle are proportional to its median lengths insome order.

Crux 2253.S24(6)376*. (T.Seimiya)

ABC is a triangle and Ib, Ic are the excentres of ABC relative to the sidesCA, AB respectively. Suppose that

IbA2 + IbC

2 = BA2 +BC2, and IcA2 + IcB

2 = CA2 + CB2.

Prove that ABC is equilateral.


Crux 2254.S24(6)377*. (T.Seimiya)

ABC is an isosceles triangle with AB = AC. Let D be the point on sideAC such that CD = 2AD. Let P be the point on the segment BD suchthat APC = 90. Prove that ABP = PCB.

Crux 2255.S24(6)378;25(2)113*. (T.Seimiya)

Let P be an arbitrary point of an equilateral triangle ABC. Prove that

| PAB − PAC| ≥ | PBC − PCB|.

Crux 2257.S24(7)427. (W.Pompe)

The diagonals AC and BD of a convex quadrilateral ABCD intersect atthe point O.Let OK, OL, OM , ON be the altitudes of triangles ABO,BCO4,CDO4, DAO respectively. Prove that if OK = OM and OL = ON ,then ABCD is a parallelogram.

Crux 2258.S24(6)380*. (W.Pompe)

In a right - angled triangle ABC (with C = 90),D lies on the segment BCso that BD = AC

√3. E lies on the segment AC and satisfies AE = CD

√3.

Find the angle between AD and BE.

Crux 2259.S24(8)509*. (Yiu)

Let X,Y,Z be the projections of the incenter of ABC on the sides BC,CAand AB respectively. Let X ′, Y ′, Z ′ be the points on the incircle diametri-cally opposite to X,Y,Z respectively. Show that the lines AX ′, BY ′, CZ ′

are concurrent.

Crux 2262.S24(7)431*. (Juan-Bosco Romero Mqrquez)

Consider two triangles ABC and A′B′C ′ such that A ≥ 90circ and A′ ≥ 90,and whose sides satisfy a > b ≥ c and a′ > b′ ≥ c′. Denote the altitude tosides a and a′ by ha and h′a. Prove that

(a) 1hah′

a≥ 1

bb′ + 1cc′ ;

(b) 1hah′

a≥ 1

bc′ + 1b′c .


Crux 2263.S24(7)432*. (T.Seimiya)

ABC is a triangle, and the internal bisectors of B, C, meet AC, AB atD, E respectively. Suppose that BDE = 30. Characterize ABC.


ABC is a right angled triangle with the right angle at A. Points D and Eare on sides AB and AC respectively such that DE//BC. Points F and Gare the feet of the perpendiculars from D, E to BC respectively.

Let I, I1, I2, I3 be the incenters of ABC, ADE, BDF , CEG,respectively. Let P be the point such that I2P//I1I3 and I3P//I1I2. Provethat the segment IP is bisected by the line BC.

Crux 2265.S24(7)433*. (W.Pompe)

Given triangle ABC, let ABX and ACY be two variable triangles con-structed outwardly on sides AB and AC of triangle ABC, such that theangles XAB and Y AC are fixed, and XBA+ Y CA = 180circ.Prove thatall the lines XY pass through a common point.

Crux 2266.S24(7)434*. (W.Pompe)

BCLK is the square constructed outwardly on side BC of an acute triangleABC. Let CD be the altitude of triangle ABC (with D on AB), and letH be the orthocenter of triangle ABC. If the lines AK and CD meet at P ,show that HP

PD = ABCD .

Crux 2267.S24(8)511. (C.Kimberling and P.Yff)

In the plane of triangle ABC, let F be the Fermat point and F ′ its isogonalconjugate. Prove that the circle through F ′ centered at A, B, C meetpairwise in the vertices of an equilateral triangle having center F .

Crux 2270.S24(7)437*. (D.J. Smeenk)

Given triangle ABC with sides a, b,c, a circle, center P , and radius ρ inte-sects sides BC, A, AB in A1, A2; B − 1, B2; C1, C2 respectively so that

A1A2

a=B1B2

b=C1C2

c= λ > 0.

Determine the locus of P .


Crux 2276.S24(8)514. (D.J. Smeenk)

Quadrilateral ABCD is cyclic with circumcircle Γ(O,R).Show that the nine - point circles of triangles BCD, CDA, DAB and

ABC have a point in common, and characterize that point.

Crux 2279.S24(8)515*. (W.Janous)

With the usual notation for a triangle, prove that∑

cyclic

sin3A cosB cosC =sr

4R4(2R2 − s2 + (2R + r)2).

In Crux 2178.S23(447), Florian Herzig showed that∑cyclic

sin3A cosB cosC =∏

sinA∑

cos2A.


ABC is a triangle with incenter I. Let D be the second intersection of AIwith the circumcircle of ABC. Let X, Y be the feet of the perpendicularsfrom I to BD, CD respectively.

Suppose that IX + IY = 12AD. Find BAC.


ABC is at triangle, and D a point on the side BC produced beyond C,such that AC = CD. Let P be the second intersection of the circumcircleof triangle ACD with the circle on diameter BC. Let E be the intersectionof BP with AC, and let F be the intersection of CP with AB.

Prove that D, E, F are collinear.

Crux 2282.S24(8)518*. (D.J. Smeenk)

A line intersects the sides BC, CA, AB of triangle ABC at D, E, Frespectively such that D is the midpoint of EF . Determine the minimumvalue of |EF | and express its length as elements of triangle ABC.


Crux 2283.S24(8)519*. (W.Pompe)

You are given triangle ABC with C = 60circ. Suppose E is an interiorpoint of line egment AC such that CE < BC. SUppose that D is an interiorpoint of line segment BC such that

AE

BD=BC

CE− 1.

Suppose that AD and BE intersect in P , and the circumcircles of AEP andBDP intersect in P and Q. Prove that QE//BC.

Crux 2284(corrected).S24(8)521*. (T.Seimiya)

ABCD is a rhombus with A = 60. Suppose that E, F are points on thesides AB, AD respectively, and that CE, CF meet BD at P , Q respectively.Suppose that BE2 +DF 2 = EF 2.Show that BP 23 +DQ2 = PQ2.

Crux 2286.

Proposed by Toshio Seimiya, Kawasaki, Japan.ABCD is a rhombus with A = 60. Suppose that E, F are points on th

sides AB, AD respectively, and that CE, CF meet BC at P , Q respectively.Suppose that BE2 +DF 2 = EF 2. Prove that BP 2 +DQ2 = PQ2.

Crux 2287=2234.S24(4)247.

Crux 2288=2251.S24(6)373.


Suppose that ABC is a triangle with sides a, b, c, that P is a point in theinterior of triangle ABC, and that AP meets the circle BPC again at A′.Define B′ and C ′ similarly.

Prove that the perimeter of the hexagon AB′CA′BC ′ satisfies

P ≥ 2(√ab+

√bc+

√ca).


Suppose that the bisector of angle A of triangle ABC intersects BC at D.Suppose that AB +AD = CD and AC +AD = BC. Determine the anglesB and C.



Suppose that ABC is a triangle with angles B and C satisfying C = 90 +12B, that the exterior bisector of angle A intersects BC at D, and that theside AB touches the incircle of triangle ABC at E. Prove that CD = 2AE.


An acute angled triangle ABC is given, and equilateral triangles ABD andACE are drawn outwardly on the sides AB and AC. Suppose that CDand BE meet AB and AC at F and G respectively, and that CD and BEintersect at P .

Suppose that the area of the quadrilateral AFPG is equal to the area oftriangel PBC. Determine angle BAC.


Suppose that ABC is a triangle and that P is a point on the circumcircle,distinct from A, B, C.Denote by SA the circle with center A and radius AP .Define SB and SC similarly. Suppose that SA and SB intersect at P andPC . Define PB and PA similarly.

Prove that PA, PB , PC are collinear.


Given triangle ABC with AB < AC. The bisectors of angles B and C meetAC and AB at D and E respectively, and DE intersects BC at F .

Suppose that DFC = 12 ( DBC − ECB). Determine angle A.


Given trinalge ABC with angles B and C satisfying C = 90+ 12B. Suppose

that M is the midpoint of BC, and that the circle with center A and radiusAM meets BC again at D. Prove that MD = AB.

Crux 2318.S25(2)123*. (V.Konecny)

Suppoe that ABC is a triangle with circumcenter O and circumradius R.Consider the bisector of any side (say AC) and let P (the pedal point) beany point on inside the circumcircle.

Let K, L, M denote the feet of the perpendicualrs from P to the linesAB, BC, CA respectively.


Show that the area [KLM ] is a decreasing function of OP .

Crux 2319.S25(2)124*. (F.Herzig)

Suppose that UV is a diameter of a semicircle, and that P , Q are two pointson the semicircle with UP < UQ. The tangents to the semicircle at P andQ meet at R. Suppose that S is the point of intersection of UP and V Q.

Prove that RS is perpendicular to UV .

Crux 2320.S25(2)126*. (D.J. Smeenk)

Two circles on the same side of the line are tangent to it at D. Thetangnets to the smaller circle from a variable point A on the large circleintersect at B andC. If b and c are the radii of the incircles of trianglesABD and ACD, prove that b+ c is independent of the choice of A.


Suppose that the ellipse E has equation

x2

a2+y2

b2= 1.

Suppose that Γ is any circle concentric with E . Suppose that A is a pointon E and B is a point on Γ such that AB is tangent to both E and Γ.

Find the maximum length of AB.

Crux 2326.S25(3)178*. (W.Janous)

Prove or disprove

2π<∑ (1 − sin A

2 )(1 + 2 sin A2 )

π −A≤ 9

2π.

Crux 2333.S25(3)187. (D.J. Smeenk)

You are given that D and E are points on the sides AC and AB respectivelyof triangle ABC. Also, DE is not parallel to CB. Suppose that F and Gare points on BC and ED respectively such that

BF : FC = EG : GD = BE : CD.

Show that GF is parallel to the angle bisector of BAC.



Suppose that ABC is a triangle with incentre I, and that BI, CI meet AC,AB at D, E respectively. Suppose that P is the intersection of AI withDE. Suppose that PD = PI. Find angle ACB.

Crux 2335.S25(3)190*. (T.Seimiya)

Triangle ABC has circumcircle Γ. A circle Γ′ is internally tangent to Γ at P ,and touches sides AB, AC at D, E respectively. Let X, Y be the feet of theperpendiculars from P to BC, DE respectively. Prove that PX = PY sinA

2 .


The bisector of angle A of a triangle ABC meets BC at D. Let Γ and Γ′

be the circumcircles of triangles ABD and ACD respectively, and let P , Qbe the intersections of AD with the common tangents to Γ, Γ′ respectively.

Prove that PQ2 = AB · AC.

Crux 2338.S25(4)243*. (Seimiya)

Suppose ABCD is a convex cyclic quadrilateral, and P is the intersectionof the diagonals AC and BD. Let I1, I2, I3, I4 be the incentres of trianglesPAB, PBC, PCD and PDA respectively. Suppose that I1, I2, I3, I4 areconcyclic.

Prove that ABCD has an incircle.

Crux 2339.S25(5)309*. (T. Seimiya)

A rhombus ABCD has incircle Γ, and Γ touches AB at T . A tangent to Γmeets sides AB, AD at P , S respectively, and the line PS meets BC, CDat Q, R respectively. Prove that

(a) 1PQ + 1

RS = 1BT ,

and(b) 1

PS − 1QR = 1

AT .

Crux 2342.S25(4)249*. (D.J. Smeenk)

Given A and B are fixed points of circle Γ. The point C moves on Γ, on oneside of AB. D and E are points outside triangle ABC such that trianglesACD and BCE are both equilateral.


(a) Show that CD and CE each pass through a fixed point of Γ whenC moves on Γ.

(b) Determine the locus of the midpoint of DE.

Crux 2346.S25(5)311. (J.B.Romero Marquez)

The angles of triangle ABC satisfy A > B > C. Suppose that H is the footof the perpendicualr from A to BC, that D is the foot of the perpendicularform H to AB, that E is the foot of the perpendicular from H to AC, thatP is the foot of the perpendicular from D to BC, and that Q is the foot ofthe perpendicular from E to AB. Prove that A is acute, right, or obtuseaccording as AH −DP −EQ is positive, zero, or negative.

Crux 2348.S25(5)312*. (D.J. Smeenk)

Without the use of trigonometrical formulae, prove that

sin 54 =12

+ sin 18.


Suppose that ABC has acute angles such that A < B < C. Prove that

sin2B sinA

2sin(A+

B

2) > sin2A sin

B

2sin(B +

A

2).

Solution: Let the bisectors of angles A and B intersect their opposite sidesat P and Q respectively. In standard notation,

CP =ab

b+ c, CQ =

ab

a+ c.

By the law of sines,

sin2A sin B2 sin(B + A

2 )sin2B sin A

2 sin(A+ B2 )

=sin2A

sin2B· sin(B + A

2 )sin A

2

· sin B2

sin(A+ B2 )

=a2

b2· CQa

· b

CP=a

b· CQCP

=a

b· b+ c

a+ c=ab+ ac

ab+ bc< 1.

since ac < bc. Since all the sines involved are positive, this results holdsunder the assumption A < B.


A slight modification of the above shows that under the same assump-tion,

sinB sinA

2sin(A+

B

2) < sinA sin

B

2sin(B +

A

2).


Suppose that the centroid of triangle ABC is G, and that M and N arethe midpoints of AC and AB respectively. Suppose that circles ANC andAMB meet at (A and) P , and that circle AMN meets AP again at T .

(a) Determine AT : AP .(b) Prove that BAG = CAT .


Determine the shape of ABC if

cosA cosB cos(A−B)+cosB cosC cos(B−C)+cosC cosA cos(C−A)+2 cosA cosB cosC = 1.


Determine the shape of ABC if

sinA sinB sin(A−B) + sinB sinC sin(B −C) + sinC sinA cos(C −A) = 0.

Crux 2354(corrected).S25(5)317. (H.Gulicher)

In triangle P1P2P3, the line joining Pi−1Pi+1 meets a line σj at the pointSi,j, (i, j = 1, 2, 3, all indices taken modulo 3), such that all the points Si,j,Pk are distinct, and different from the vertices of the triangle.

(1) Prove that if all the point Si,i are non - collinear, then any two ofthe following conditions imply the third condition:

(a) P1S3,1

S3,2P2· P2S1,2

S1,2P3· P3S2,3

S2,3P1= −1;

(b) S1,2S1,1

S1,1S1,3· S2,3S2,2

S2,2S2,1· S3,1S3,3

S3,3S3,2= 1;

(c) σ1, σ2, σ3 are either concurrent or parallel.

(2) Prove further that (a) and (b) are equivalent if the Si,i are collinear.


Crux 2355.S25(5)318. (G.P.Henderson)

For j = 1, 2, . . . ,m, let Aj be non-collinear points with Aj = Aj+1. Translateevery even-numbered point by an equal amount to get new points A′

2, A′4,

. . . , and consider the sequence Bj , where B2i = A′2i and B2i−1 = A2i−1.

The laast member of the new sequence is either Am+1 or A′m+ 1 accordingas m is even or odd.

Find a necessary and sufficient condition for the length of the pathB1B2B3 · · ·Bm to be greater than the length of the path A1A2A3 · · ·Am

for all such nonzero translations.Crux 1985 provides an example of such a configuration. There, m = 2n,

the Ai are the vertices of a regular 2n−gon, and A2n+1 = A1.

Crux 2356.S25(6)369. (V.Oxman)

Five points, A, B, C, K, L, with whole number coordinates are given. Thepoints A, B, C do not lie on a line. Prove that it is possible to find twopoints, M and N , with whole number coordinates, such that M lies on theline KL and KLM is similar to ABC.

Crux 2358.S25(6)371*. (G.Leversha)

In triangle ABC, let the midpoints of BC, CA, AB be L, M , N respectively,and let the feet of the altitudes from A, B, C be D, E, F respectively. LetX be the intersection of LE and MD, let Y be the intersection of MF andNE, and let Z be the intersection of ND and LF . Show that X, Y , Z arecollinear.

Solution: We use homogeneous barycentric coordinates with respect to tri-angle ABC, and interchange the labelling of the points X and Z. For con-venience, write a′ := b2 + c2 − a2, b′ := c2 + a2 − b2, and c′ = a2 + b2 − c2.Since

BD : DC = c cosB : b cosC = c2 + a2 − b2 : a2 + b2 − c2 = b′ : c′,

D is the point with homogeneous coordinates 0 : c′ : b′. Since the midpointM of CA has homogeneous coordinates 1 : 0 : 1, the equation of the lineDM is

det

⎛⎝x y z

1 0 10 c′ b′

⎞⎠ = 0,

orc′x+ b′y − c′z = 0.


By interchanging a, b, and x, y, we obtain the equation of LE:

a′x+ c′y − c′z = 0.

It is easy to find the intersection of LE and MD as the point

Z = c′(c′ − b′) : c′(c′ − a′) : c′2 − a′b′.

Similarly,Y = b′(b′ − c′) : b′2 − c′a′ : b′(b′ − a′).

andX = a′2 − b′c′ : a′(a′ − c′) : a′(a′ − b′).

Now, these points X, Y , Z are collinear since the determinant

det

⎛⎝ a′2 − b′c′ a′(a′ − c′) a′(a′ − b′)b′(b′ − c′) b′2 − c′a′ b′(b′ − a′)c′(c′ − b′) c′(c′ − a′) c′2 − a′b′

⎞⎠ = 0.

The equation of the line is

a′(b′ + c′)(b′ − c′)2x+ b′(c′ + a′)(c′ − a′)2y + c′(a′ + b′)(a′ − b′)2z = 0.

In terms of a, b, c, this is

a2(b2−c2)2(b2+c2−a2)x+b2(c2−a2)(c2+a2−b2)y+c2(a2−b2)(a2+b2−c2)z = 0.

Alternative solution: The points D, E, F , L, M , N are concyclic, all lyingon the nine - point circle. The collinearity of X, Y , Z follows immediatelyfrom Pascal’s mystic hexagram theorem. See, for example, Pedoe, Geometry,Dover reprint, 1988, p.335.

Crux 2359.S25(6)372. (V.N.Murty)

Let PQRS be a parallelogram. Let Z divide PQ internally in the ratio k : .The line through Z parallel to PS meets the diagonal SQ at X. The lineZR meets SQ at Y . Find the ratio XY : SQ.


In triangle ABC, let BE and CF be internal angle bisectors, and let BQand CR be altitudes, where F and R lie on AB, and Q and E lie on AC.Assume that E, Q, F , and R lie on a circle that is tangent to BC.


Prove that triangle ABC is equilateral.

Solution: The lengths of the various segments are

AR = b cosA, AQ = c cosA, AF =bc

a+ b, AE =

bc

a+ c.

Since E, Q, F , and R lie on a circle, AE · AQ = AF · AR, i.e., ca+c = b

a+b .From this, we have b = c.

Now, CQ = a cosC, and CE = aba+c = ac

a+c . Since this circle is tangent toBC (necessarily at its midpoint), CQ ·CE = (a

2 )2. From this, cosC = a+c4c .

Since cosC = a2c , we have a = c, and the triangle is equilateral.

Crux 2361.S25(6)374*. (K.R.S.Sastry)

The lengths of the sides of triangle ABC are given by relatively prime naturalnumbers. Let F be the point of tangency of the incircle with side AB.Suppose that ABC = 60 and AC = CF . Determine the lengths of thesides of triangle ABC.

Solution: Let s = (a+ b+ c)/2. Note that BF = s− b. Since ABC = 60,we have from the triangles ABC and FBC,

b2 = a2 − ac+ c2, (3)b2 = a2 − a(s− b) + (s− b)2. (4)

Upon subtraction, we obtain

0 =12(s − a)(a+ b− 3c).

Since s− a = 0, we must have b = 3c− a. From (1), we have c(5a− 8c) = 0.Since the sides are relative prime natural numbers, a = 8, c = 5, and b = 7.

Crux 2365.S25(6)379. (V.Oxman)

Triangle DAC is equilateral. B is on the line DC so that BAC = 70. Eis on the line AB so that ECA = 55. K is the midpoint of ED. Withoutthe use of a computer, calculator, or protractor, show that 60 > AKC >57.5.


Crux 2366.S25(6)380*. (C.Shevlin)

Triangle ABC has area p, where p ∈ N. Let

Σ = min(AB2 +BC2 + CA2)

where the minimum is taken over all possible triangles ABC with area p,and where Σ ∈ N.

Find the least value of p such that Σ = p2.

Crux 2367.S25(6)381*. (K.R.S.Sastry)

In triangle ABC, the cevians AD, BE intersect at P . Prove that

[ABC] × [DPE] = [APB] × [CDE].

Here, [ABC] denotes the area of triangle ABC etc.

Crux 2375.S25(7)436*. (T.Seimiya)

Let D be a point on side AC of triangle ABC. Let E and F be points onthe segments BD and BC, respectively, such that BAE = CAF . LetP and Q be points on BC and BD respectively, such that EP//DC andFQ//CD. Prove that BAP = CAQ.

Crux 2376.S25(7)437*. (A.White)

Suppose that ABC is a right angled triangle with the right angle at C.Let D be a point on hypotenuse AB, and leet M be the midpoint of CD.Suppose that AMD = BMD. Prove that

(1) AC2MC2 + 4[ABC][BCD] = AC2MB2;(2) 4AC2MC2 −AC2BD2 = 4[ACD]2 − 4[BCD]2.

Crux 2377.S25(7)438. (N.Dergiades)

Let ABC be a triangle and P a point inside it. Let BC = a, CA = b,AB = c, PA = x, PB = y, PC = z, BPC = α, CPA = β and APB = γ.

Prove that ax = by = cz if and only if α−A = β −B = γ − C = π3 .



Suppose that M1, M2, M3 are the midpoints of the altitudes from A to BC,from B to CA and from C to AB in ABC. Suppose that T1, T2, T3 arethe points where the excircles to ABC opposite A, B, and C touch BC,CA, and AB.

Prove that M1T1, M2T2 and M3T3 are concurrent.Determine the point of concurrency.

Solution: These lines are concurrent at the incenter I of the triangle, whichhas barycentric coordinate 1

2s [aA+ bB + cC].Since BT1 : T1C = (s− c) : (s − b),

T1 =1a[(s− b)B + (s − c)C].

Let P be the projection of A on BC. Since

BP : PC = c cosB : b cosC = c2 + a2 − b2 : a2 + b2 − c2,

P is the point 12a2 [(a2 + b2 − c2)B + (c2 + a2 − b2)C], and the midpoint of

the altitude AP is

M1 =1

4a2[2a2A+ (a2 + b2 − c2)B + (c2 + a2 − b2)C]

=1

4a2[2a2A+ 2abB + 2acC − 4(s − a)(s− b)B − 4(s − a)(s− c)C]

=12a

(aA+ bB + cC) − s− a

aT1

=1a[sI − (s− a)T1].

In other words,

I =1s[aM1 + (s− a)T1].

The line M1T1 therefore contains the incenter of the triangle; so do the linesM2T2 and M3T3.

Crux 2382.S25(7)440*. (M.Aassila)

If ABC has inradius r and circumradius R, show that

cos2 B − C

2≥ 2rR.


Solution: We shall assume B ≤ C, so that the inequality is equivalent to

2R sin(B +A

2) = 2R cos

C −B

2≥ 2

√2Rr.

Suppose the bisector of angle A intersects the circumcircle at M . Notethat AM = 2R sin(B + A

2 ). Also, IM = BM = 2R sin A2 , and AI = r

sin A2

,

where I is the incenter. Consequently,

2R sin(B +A

2) = AM = AI + IM ≥ 2

√AI · IM = 2

√2Rr.

This completes the proof.

Remark. Equality holds if and only if I is the midpoint of AM . This is thecase if and only if tan B

2 tan C2 = 1

3 .

Crux 2383.S25(7)441. (M.Aassila)

Suppose that three circles, each of radius 1, pass through the same point inthe plane. Let A be the set of points which lie inside at least two of thecircles. What is the least area that A can have?

See also Crux 2483.

Crux 2397. (T.Seimiya)

Given a right - angled triangle ABC with BAC = 90. Let I be theincenter, and let D and E be the intersections of BI and CI with AC andAB respectively. Prove that BI2+ID2

CI2+IE2 = AB2

AC2 .

Solution: Let r be the inradius of the triangle. Then, BI = rsin B

2

and

ID = rcos B

2

. It follows that

BI2 + ID2 = r2(1

cos2 B2

+1

sin2 B2

) =4r2

sin2B.

Similarly, CI2 + IE2 = 4r2

sin2 C, and the result follows from the sine law.


Given a square ABCD with points E and F on sides BC and CD respec-tively, let P and Q be the feet of the perpendiculars from C to AE and AFrespectively. Suppose that CP

AE + CQAF = 1. Prove that EAF = 45.


Solution: Suppose the square has unit side length. If CAP = θ < 45,then

CP

AE=

√2 sin θ cos(45 − θ) =

12

+1√2

sin(45 − 2θ).

Similarly, if CAQ = φ < 45, then

CQ

AF=

12

+1√2

sin(45 − 2φ).

It follows that CPAE + CQ

AF = 1 if and only if sin(45−2θ)+sin(45−2φ) = 0.This is possible only when (45 − 2θ) + (45 − 2φ) = 0, i.e., θ + φ = 45.

Crux 2407. (C.J.Bradley)

Triangle ABC is given with BAC = 72. The perpendicular from B toCA meets the internal bisector of BCA at P . The perpendicular from Cto AB meets the internal bisector of ABC at Q.

If A, P , Q are collinear, determine ABC and BCA.

Solution: These angles are 84 and 24.We begin by considering a generic triangle ABC, with sides a, b, c,

and opposite angles α, β, γ respectively. The projection of B on CA hasbarycentric coordinates

E =1b((a cos γ)A+ (c cosα)C).

The bisector of angle C meets BE at

P =1

1 + cos γ(E+(cos γ)B) =

1b(1 + cos γ)

((a cos γ)A+(b cos γ)B+(c cosα)C).

Similarly, the bisector of angle B intersects the altitude from C at

Q =1

c(1 + cos β)((a cos β)A+ (b cosα)B + (c cos β)C).

The points A, P , Q are collinear if and only if

det

⎛⎝ 1 0 0a cos γ b cos γ c cosαa cos β b cosα c cos β

⎞⎠ = 0.

This reduces to cos β cos γ = cos2 α. From this,


cos(β−γ) = 2 cos β cos γ− cos(β+γ) = 2 cos2 α+cosα = 1+cos 2α+cosα.

Now, with α = 72, we have

cos(β − γ) = 1 + cos 144 + cos 72 = 1 − sin 54 + sin 18 =12,

by Crux 2348. It follows that β − γ = 60. Since β + γ = 108, we haveβ = 84 and γ = 24.

This completes the proof.

Remarks. (1) From the above calculation, it follows that a unique triangleABC exists with A, P , Q collinear for every acute angle α greater than 60.

(2) Consider the counterparts of the line PQ for the other two pairs ofvertices A, C, and A, B, we obtain three lines. These three lines are alwaysconcurrent, and the intersection is the point K on the line OI such thatOI : IK = R : r. Here, O and I are respectively the circumcenter andincenter, and R, r the circumradius and inradius of the triangle.

I omit the details, and perhaps shall propose it as a separate problem.

Crux 2415. (P.Yiu)

Given a point Z on a line segment AB, find a euclidean construction of aright - angled triangle ABC whose incircle touches hypotenuse AB at Z.

Crux 2416. (V.Konecny)



In triangle ABC, the lengths of the sides BC, CA, AB are 1998, 2000, 2002respectively. Prove that there exists exactly one point P (distinct from Aand B) on the minor arc AB of the circumcircle of triangle ABC such thatPA, PB, PC are all of integer length.

Crux 2422. (W.Janous)

Let A, B, C be the angles of an arbitrary triangle. Prove or disprove that

1A

+1B

+1C

≥ 9√

3

2π(sinA sinB sinC)13

.


Crux 2424. (K.R.S.Sastry)

In triangle ABC, suppose that I is the incenter and BE is the bisectorof angle ABC, with E on AC. Suppsoe that P is on AB and Q on ACsuch that PIQ is parallel to BC. Prove that BE = PQ if and only if ABC = 2 ACB.


Suppose that D is the foot of the altitude from vertex A of an acute angledHeronian triangle. Suppose that the greatest common divisor of the sidelengths is 1. Find the smalelst possible value of the side length BC, giventhat BD −DC = 6.





Crux 2431. (J.Taylor)




Crux 2437. (Yiu)

Let P be a point in the plane of triangle ABC. If the midpoints of thesegments AP , BP , CP all lie on the nine-point circle of triangle ABC,must P be the orthocenter of this triangle ?

Solution: We use barycentric coordinates with respect to ABC. Denote themidpoints of AP , BP , CP by X, Y , Z respectively. The triangles XY Zand DEF are homothetic, the center of homothety, being the midpoint ofDX, is the point K = 3G+P

4 . The circumcenters of these two triangles aresymmetric with respect to the center of homothety. If these triangles havethe same circumcircle, then K is their common circumcenter. It followsthat K is the nine-point center, midway between the orthocenter and thecircumcenter, or 3G+H

4 , by the Euler line theorem. From this P = H, theorthocenter.


Crux 2438. (P.Hurthig)

Show how to tile an equilateral triangle with congruent pentagons. Reflec-tions are allowd. Compare Crux 1988.

Crux 2454. (G. Leversha)

Three circles intersect each other orthogonally at pairs of points A and A′.B and B′, and C and C ′. Prove that the circumcircles of triangles ABCand AB′C ′ touch at A.


Three equal circles, centered at A, B. and C, intersect at a common pointP . The other intersection points are L (not on the circle center A), M (noton the circle center B), and N (not on the circle center C). Suppose that Qis the centroid of triangle LMN , that R is the centroid of ABC, and thatS is the circumcenter of LMN .

(a) Show that P , Q, R are collinear.(b) Establish how they are distributed on the line.


Two circles intersect orthogonally at P . A third circle touches them at Qand R. Let X be any point on this third circle. Prove that the circumcirclesof triangle XPQ and XPR intersect at 45.


In quadrilateral ABCD, we have A + B = 2α < 180,and BC = AD.Construct isosceles triangles DCI, ACJ , and DBK, where I, J , K are onthe other side of CD from A, such that

ICD = IDC = JAC = JCA = KDB = KBD = α.

(a) Show that I, J , K are collinear.(b) Establish how they are distributed on the line.

Crux 2458. (N. Dergiades)

Let ABCD be a quadrilateral inscribed in the circle centre O, radius R, andlet E be the point of intersection of the diagonals AC and BD. Let P be


any point on the line segment OE and let K, L, M , N be the projectionsof P on AB, BC, CD, DA respectively.

Prove that the lines KL, MN , AC are either parallel or concurrent.

Crux 2462. (V.N.Murty)

If the angles A, B, C of triangle ABC satisfy

cosA sinA

2= sin

B

2sin

C

2,

prove that triangle ABC is isosceles.

Crux 2464. (M.Lambrou)

Given triangle ABC with circumcircle Γ, the circle ΓA touches AB and ACat D1 and D2, and touches |Gamma internally at L. Define E1, E2, M , andF1, F2, N in a corresponding way. Prove that

(a) AL, BM , CN are concurrent;(b) D1D2, E1E2, F1F2 are concurrent, and that the point of concurrency

is the incenter of triangle ABC.

Crux 2466. (V.Oxman)

Given a circle (but not its center) and two of its arcs, AB and CD, and theirmidpoints M and N (which do not coincide and are not the end points of adiameter), prove that all the unmarked straightedge and compass construc-tion that can be carried out in the plane of the circle can also be done withan unmarked straightedge alone.


Givne a line segment UV and two rays r and s, emanating from V such thatangle(UV, r) = (r, s) = 60 and two lines g, h on U such that (UV, g) = (g, h) = α, where 0 < α < 60. The quadrilateral ABCD is determinedby g, h, r, s. Let P be the point of intersection of AB and CD. Determinethe locus of P as α varies from 0 to 60.


Crux 2469. (P.Yiu)

Crux 2470. (P.Yiu)

Crux 2473. (M.A.Covas)

Given a point S on the side AC of triangle ABC, construct a line through Swhich cuts lines BC and AB at P and Q respectively, such that PQ = PQ.


Given a nondegenerate triangle ABC with circumcircle Γ, let rA be theinradius of rhe region bounded by BA, AC and arc (CB) (s0 that the regionincludes the triangle). Similarly, define rB and rC . As usual, r and R arethe inradius and circumradius of triangle ABC. Prove that

(a) 6427r

3 ≤ rArBrC ≤ 3227Rr

2;(b) 16

3 r2 ≤ rBrC + rCrA + rArB ≤ 8

3Rr;(c) 4r ≤ rA + rB + rc ≤ 4

3(R+ r).

Crux 2483. (V.Konecny)

Suppose that 0 ≤ A,B,C, and A+B + C ≤ π. Show that

0 ≤ A− sinA− sinB − sinC + sin(A+B) + sin(A+ C) ≤ π.

There are, of course, similar inequalities with the angles permuted cyclically.See Crux 2383.


Given a square ABCD, suppose that E is a point on AB produced beyondB, that F is a point on AD [rodiced beyond D, and that EF = 2AB. LetP and Q be the intersections of EF with BC and CD respectively. Provethat

(a) APQ is acute angled;(b) PAQ ≥ 45.


ABCD is a convex quadrilateral with AB = BC = CD. Let P be theintersection of the diagonals AC and BD. Suppose that AP : BD = DP :AC. Prove that either BC//AD or AB ⊥ CD.


Crux 2486. (J.Howard)

It is well known that cos 20 cos 40 cos 80 = 18 . Show that

sin 20 sin 40 sin 80 =√

38.

Crux 2887 (V. N. Murty)

If a, b, c are the sides of triangle ABC in which at most one angle exceedsπ3 , and if R is its circumradius, prove that

a2 + b2+2 ≤ 6R2∑

cyclic

cosA.

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