UCD Mathematical Enrichment 10/03/18 Geometry: Similar Triangles Warm-up problems 1. Let AP be tangent to the circle K at P and let B,C lie on K. Prove that | 6 AP B| = | 6 PCB| 2. Let P be a point not on the circle K and let the lines l 1 ,l 2 pass through P and cross K at A 1 ,B 1 and A 2 ,B 2 respectively. Using similar triangles, prove that |PA 1 ||PB 1 | = |PA 2 ||PB 2 | 3. Let the circles K 1 and K 2 intersect at A and B. Let P be a point on AB and let l 1 ,l 2 be two lines through P . Suppose l 1 intersects K 1 at Q, R and l 2 intersects K 2 at S, T . Prove that QRST is a cyclic quadrilateral. (Hint: Use the result from the problem 2.) Intermediate problems 4. (IrMO 2013) The altitudes of a triangle 4ABC are used to form the sides of a second triangle 4A 1 B 1 C 1 . The altitudes of 4A 1 B 1 C 1 are then used to form the sides of a third triangle 4A 2 B 2 C 2 . Prove that 4A 2 B 2 C 2 is similar to 4ABC . 5. (IrMO 2014) The square ABCD is inscribed in a circle with centre O. Let E be the midpoint of AD. The line CE meets the circle again at F . The lines FB and AD meet at H . Prove |HD| =2|AH |. 6. (BMO 2005, Round 1) Let 4ABC be an acute-angled triangle, and let D,E be the feet of the perpendiculars from A, B to BC,CA respectively. Let P be the point where the line AD meets the semicircle constructed outwardly on BC , and Q be the point where the line BE meets the semicircle constructed outwardly on AC . Prove that |CP | = |CQ|. Advanced problems 7. (IMO 2014) Points P and Q lie on side BC of acute-angled triangle 4ABC so that | 6 P AB| = | 6 BCA| and | 6 CAQ| = | 6 ABC |. Points M and N lie on lines AP and AQ respectively, such that P is the midpoint of AM and Q is the midpoint of AN . Prove that the lines BM and CN intersect on the circumcircle of 4ABC . 8. (IMO 2017) Let R and S be different points on a circle Ω such that RS is not a diameter. Let l be the tangent line to Ω at R. Point T is such that S is the midpoint of the line segment RT . Point J is chosen on the shorter arc RS of Ω so that the circumcircle Γ of triangle 4JST intersects l at two distinct points. Let A be the common point of Γ and l that is closer to R. Line AJ meets Ω again at K. Prove that the line KT is tangent to Γ.
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UCD Mathematical Enrichment 10/03/18
Geometry: Similar Triangles
Warm-up problems
1. Let AP be tangent to the circle K at P and let B,C lie on K. Prove that
|6 APB| = | 6 PCB|
2. Let P be a point not on the circle K and let the lines l1, l2 pass through P and cross K at A1, B1
and A2, B2 respectively. Using similar triangles, prove that
|PA1||PB1| = |PA2||PB2|
3. Let the circles K1 and K2 intersect at A and B. Let P be a point on AB and let l1, l2 be two linesthrough P . Suppose l1 intersects K1 at Q,R and l2 intersects K2 at S, T . Prove that QRST is acyclic quadrilateral. (Hint: Use the result from the problem 2.)
Intermediate problems
4. (IrMO 2013) The altitudes of a triangle 4ABC are used to form the sides of a second triangle4A1B1C1. The altitudes of 4A1B1C1 are then used to form the sides of a third triangle 4A2B2C2.Prove that 4A2B2C2 is similar to 4ABC.
5. (IrMO 2014) The square ABCD is inscribed in a circle with centre O. Let E be the midpoint of AD.The line CE meets the circle again at F . The lines FB and AD meet at H. Prove |HD| = 2|AH|.
6. (BMO 2005, Round 1) Let 4ABC be an acute-angled triangle, and let D,E be the feet of theperpendiculars from A,B to BC,CA respectively. Let P be the point where the line AD meets thesemicircle constructed outwardly on BC, and Q be the point where the line BE meets the semicircleconstructed outwardly on AC. Prove that |CP | = |CQ|.
Advanced problems
7. (IMO 2014) Points P and Q lie on side BC of acute-angled triangle 4ABC so that | 6 PAB| =|6 BCA| and | 6 CAQ| = | 6 ABC|. Points M and N lie on lines AP and AQ respectively, such thatP is the midpoint of AM and Q is the midpoint of AN . Prove that the lines BM and CN intersecton the circumcircle of 4ABC.
8. (IMO 2017) Let R and S be different points on a circle Ω such that RS is not a diameter. Let l bethe tangent line to Ω at R. Point T is such that S is the midpoint of the line segment RT . PointJ is chosen on the shorter arc RS of Ω so that the circumcircle Γ of triangle 4JST intersects l attwo distinct points. Let A be the common point of Γ and l that is closer to R. Line AJ meets Ωagain at K. Prove that the line KT is tangent to Γ.
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