03 Advanced Problem Solving Geometry - IMSA

Advanced Problem Solving Geometry Fall 2013 1) The figure shows right triangle ΔABC together with points P and Q such that the lengths PC = BC and QA = BA. If the inradius of ΔABC is 10, compute the circumradius of ΔPBQ.

Name: ________________________ A P

Q

B

C

2) In ΔABC with circumcenter O, orthocenter H, and circumradius R, prove: a) m∠HAO = | m∠B – m∠C| b) OH2 = 9R2 – a2 – b2 – c2

  AB DE meet at X, 3) Let A, B, C, D, E, and F be any six points on a circle. Let the lines and     BC and EF meet at Y, and FA and CD meet at Z. Prove that X, Y, and Z are collinear. (This is actually true of any conic section, not just circles.)