Problem

Source: Iran Third Round 2013 - Geometry Exam - Problem 5

Tags: geometry, circumcircle, geometric transformation, reflection, perpendicular bisector, geometry unsolved



Let $ABC$ be triangle with circumcircle $(O)$. Let $AO$ cut $(O)$ again at $A'$. Perpendicular bisector of $OA'$ cut $BC$ at $P_A$. $P_B,P_C$ define similarly. Prove that : I) Point $P_A,P_B,P_C$ are collinear. II ) Prove that the distance of $O$ from this line is equal to $\frac {R}{2}$ where $R$ is the radius of the circumcircle.