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Solve First – ask questions laterPhilip Todd,
Saltire Software
Please turn your smart phones ON…and browse to this
location:
(If, like me, your phone sits on your desk at home, don’t
worry.. These are supplemental interactive materials and not
essential to the presentation.)
Browse to …
http://goo.gl/D14OQL
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Solve First – ask questions laterPhilip Todd,
Saltire Software
Please turn your smart phones ON…and browse to this
location:
(If, like me, your phone sits on your desk at home, don’t
worry.. These are supplemental interactive materials and not
essential to the presentation.)
Browse to …
http://goo.gl/D14OQL
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-
Problem Solving Paradigms
Problem SolutionModelBy hand computation
Mathematical insight
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Problem Solving Paradigms
Problem SolutionModelBy hand computation
Problem Solution
Brute force computation
Mathematical insight
Mathematical insight
Model
Exploration of solution space
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MuPad
Derive
TI nSpire
Maxima
ClassPad Manager
… (MathML)http://goo.gl/D14OQL
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South Bohemia Mathematical Letters Volume 22, (2014), No. 1,
26–37.
THE SIMSON-WALLACE LOCUS IN PLANE AND SPACE
PAVEL PECH1, EMIL SKRˇ´ISOVSKˇ Y´2
Abstract. In the paper several theorems related to the
well-known Simson– Wallace theorem are given. Some properties of
the nine-point circle and circumcircle of a given triangle are
investigated. Further the relation between two Simson lines is
studied, obtaining Half Angle Theorem. Special attention is paid to
Steiner Deltoid curve as the envelope of the system of
Simson-Wallace lines whose equation was derived. Simultaneously the
generalization of the theorem into space is described and further
examined.
1. Introduction
The Simson–Wallace theorem describes an interesting property
regarding the points on the circumcircle of the triangle [1, 3, 4,
9].
Theorem 1.1 (Simson-Wallace). Let ABC be a triangle and P a
point in plane. The feet of the perpendicular lines onto the sides
of the triangle are collinear if and only if P lies on the
circumcircle of ABC, Fig. 1.
Figure 1. The Simson-Wallace Theorem – points T,U,V are
collinear.
Proof. To prove the theorem, we use analytical methods. We adopt
a coordinate system where A = [0,0], B = [b,0] and C = [c1,c2],
Fig. 2. We denote the three
Received by the editors . 1991 Mathematics Subject
Classification. Key words and phrases. Simson-Walace theorem,
family of Simson lines, Steiner deltoid, Cayley’s cubic.
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Theorem 1.1 (Simson-Wallace). Let ABC be a triangle and P a
point in plane. The feet of the perpendicular lines onto the sides
of the triangle are collinear if and only if P lies on the
circumcircle of ABC, Fig. 1.
Figure 1. The Simson-Wallace Theorem – points T,U,V are
collinear.
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Theorem 1.2 (Gergonne). Let ABC be a triangle and P a point in
plane. If P lies on a circle concentric to the circumcircle, then
the area of the pedal triangle TUV is constant, Fig. 3.
Corollary 1.4. The pedal triangle degenerates into the line if
and only if P lies on the circumcircle.
Figure 3. Gergonne’s generalization – area of TUV is
constant.
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Project Characteristics
Problems I did not know the answer to.Relevance inside
mathematics.Relevance outside of mathematics.Discover new
mathematics.
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Discovering a New World?
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Discovering a New World?
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Discovering New Mathematics?
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Discovering New Mathematics?
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Keep your eyes open for…
3 succinct geometry problems
1 nice calculus problem
differential geometry sneaking in while we aren’t looking
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2006: Pedal Triangles Circle limits
2008: Telescope aberration
2009: Solar cookersCentral pivot irrigation
2010: Savonius wind turbinesBending soccer kicksCatenary
reflectors
2011: Chaotic dynamic systems
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What happens to the circumcircle of a triangle when the three
points coalesce?
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X=f(T)
Y=g(T)
A
B
C
D
t
-h+t
h+t
radius Þ (-f(t)+f(h+t))2+(-g(t)+g(h+t))2 ·
(-f(t)+f(-h+t))2+(-g(t)+g(-h+t))2 ·
(-f(h+t)+f(-h+t))2+(-g(h+t)+g(-h+t))2
2·(-f(h+t)·g(t)+f(-h+t)·g(t)+f(t)·g(h+t)-f(-h+t)·g(h+t)-f(t)·g(-h+t)+f(h+t)·g(-h+t))
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X=f(T)
Y=g(T)
A
B
Ct
-h+t
h+t
D
E
F
G
r9Þ
-f(t)2
+f(h+t)
2
2
+-g(t)
2+
g(h+t)2
2
·-f(t)2
+f(-h+t)
2
2
+-g(t)
2+
g(-h+t)2
2
·-f(h+t)
2+
f(-h+t)2
2
+-g(h+t)
2+
g(-h+t)2
2
-f(h+t)·g(t)2
+f(-h+t)·g(t)
2+
f(t)·g(h+t)2
-f(-h+t)·g(h+t)
2-
f(t)·g(-h+t)2
+f(h+t)·g(-h+t)
2
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Anticomplentary Circle:
( ) RR 22lim = Apollonius Circle:
The radius of the Circle of Apollonius is not directly related
to the circumradius.
Bevan Circle:
( ) RR 22lim = Brocard Circle:
2/)cos(2
)(sin41lim
2
RR
=
−ω
ω
Conway Circle:
( ) 0lim 22 =+ sr Cosine Circle:
( ) 0)tan(lim =ωR De Longchamps Circle:
( ) RCBAR 4coscoscos4lim =− Excircles:
( ) RCBAR 4)2/cos()2/cos()2/sin(4lim = Note that this equation
only applies to the excircle opposite angle A. The excircles
opposite angles B and C have radii of 0 because sin(0)=0
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->0 radius 20
-> infinite radius 2
-> constant multiple of radius of curvature
22
-> some other radius 6
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2006: Pedal Triangles Circle limits
2008: Telescope aberration
2009: Solar cookersCentral pivot irrigation
2010: Savonius wind turbinesBending soccer kicksCatenary
reflectors
2011: Chaotic dynamic systems
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Q: Use your calculation for angle of entry in the model to find
the focal surface
A: In figure 6, a trace on the reflected beam shows 5 optical
focal points where incoming light is angled at approximately ±9,
±4.5 and 0 degrees. Students may find that the mirror produces a
focal surface that is curved
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Q1: How do you define “optical focus point”
Q2: What is that curved surface?
Q3 Can we define “aberration”geometrically
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The parabolic mirror has no aberration at the center and
gradually increasing aberration towards the edge.
Can we change the geometry to spread the aberration more
evenly?
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X=f(T)
Y=g(T)
AC
th+t
D
E
F
z0 Þf(t)2-2·f(t)·f(h+t)+f(h+t)2+g(t)2-2·g(t)·g(h+t)+g(h+t)2 ·
f'(t)2+g'(t)2 · f'(h+t)2+g'(h+t)2
4·(f'(t)·f'(h+t)+g'(t)·g'(h+t))·(-f'(h+t)·g'(t)+f'(t)·g'(h+t))
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2006: Pedal Triangles Circle limits
2008: Telescope aberration
2009: Solar cookersCentral pivot irrigation
2010: Savonius wind turbinesBending soccer kicksCatenary
reflectors
2011: Chaotic dynamic systems
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1. What angle should the lid be opened to?
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Solar Concentration Ratio
D
d
Solar Concentration Ratio = Dd
2
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2. What is the maximum solar concentration achievable with this
box?
What angle would the lid be at when the maximum is achieved?
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Box Solar Cooker 2
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θ
AB
C Þd· 2-2·cos(θ) ·sin(θ)
| 2·sin(θ)-sin(2·θ)
1-cos(θ)>0
d
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> sin(theta)*(2-2*cos(theta))^(1/2)*d;
> diff(%,theta);
> solve(%=0,theta);
>
( )sin θ − 2 2 ( )cos θ d
+ ( )cos θ − 2 2 ( )cos θ d ( )sin θ2 d
− 2 2 ( )cos θ
,− + ( )arctan 2 2 π − ( )arctan 2 2 π
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π-arctan 2· 2
AB
C Þ8· 3·d
9
d
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AB
C
1
x
Þx2· 4-x2
2
| 4-x2 >0
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A little calculus
642
22
44
xxdxxd
−=
−⋅=
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A little calculus
642
22
44
xxdxxd
−=
−⋅=
differentiate
( )2353
382616
xxxx−=
−
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AB
C
1
2· 2· 33
Þ8· 3
9
Þ π-arccos13
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Tetrahedral Angle
31cos 1−=CEF
31cos 1−−= πDEC
5.109≈
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Focal Property of the parabola
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Converse?
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Converse
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What shape should you use for a solar cooker if you cannot
reposition it to point at the sun?
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F number
f
D
F number = fD
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High F number
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Medium F number
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Low F number
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3. What F number needs repositioned least often?
(a) small
(b) medium(c) large
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arctan 16 r f + + − d4 32 f2 d2 256 f4 256 r2 f2
> simplify(diff(%,f)); 16 ( ) − d2 16 f2 r
( ) + d2 16 f2 + + − d4 32 f2 d2 256 f4 256 r2 f2
> solve(%=0,f);
,d4 −d4
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And the winner is…
(b) medium
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2006: Pedal Triangles Circle limits
2008: Telescope aberration
2009: Solar cookersCentral pivot irrigation
2010: Savonius wind turbinesBending soccer kicksCatenary
reflectors
2011: Chaotic dynamic systems
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-
How would a catenary compare with a parabola as a solar cooker
shape?
Where would the focus be?
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2011: Chaotic dynamic systems
2012: Interactive Euclid’s Elements Books 1-4 iBook
2013: Interactive Euclid’s Elements Books 1-6 iBook
2014: Interactive Atlas of the 4 Bar Linkage iBook
2015: Printamotion (www.printamotion.com)
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94. Regiomontanus’Maximum Problem
At what point on the earth’s surface does a perpendicularly
suspended rod appear longest?
Johann Müller Regiomontanus1436-1476
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Isaac Newton1642-1727
Johann Müller Regiomontanus1436-1476
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Student work
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Browse to: Explore>Student Projects
and Explore>Journal of Symbolic Geometry
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Workshopfrom inductive reasoning to mathematical induction
PC laboratory 16:00
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Solve First – ask questions later�Snímek číslo 2Snímek číslo
3Snímek číslo 4Solve First – ask questions later�Problem Solving
ParadigmsProblem Solving ParadigmsSnímek číslo 8Snímek číslo
9Snímek číslo 10Snímek číslo 11Snímek číslo 12Snímek číslo 13Snímek
číslo 14Snímek číslo 15Snímek číslo 16Snímek číslo 17Snímek číslo
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38Discovering a New World?Discovering a New World?Discovering New
Mathematics?Discovering New Mathematics?Keep your eyes open
for…Snímek číslo 44Snímek číslo 45Snímek číslo 46Snímek číslo
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95Snímek číslo 96Snímek číslo 97Snímek číslo 98Snímek číslo
99Snímek číslo 100Snímek číslo 101Solar Concentration RatioSnímek
číslo 103Snímek číslo 104Snímek číslo 105Snímek číslo 106Snímek
číslo 107Snímek číslo 108A little calculusA little calculusSnímek
číslo 111Tetrahedral AngleSnímek číslo 113Snímek číslo 114Snímek
číslo 115Snímek číslo 116Focal Property of the
parabola�Converse?�Converse�Snímek číslo 120F number�High F
number�Medium F number�Low F number�Snímek číslo 125Snímek číslo
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146Snímek číslo 14794. Regiomontanus’ Maximum Problem Snímek číslo
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workSnímek číslo 166Snímek číslo 167Snímek číslo 168Snímek číslo
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