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Collinear Point On Line With Distance R/2 To Circumcenter

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

September 8, 2013
geometrycircumcirclegeometric transformationreflectionperpendicular bisectorgeometry unsolved

Problem Statement

Let ABCABC be triangle with circumcircle (O)(O). Let AOAO cut (O)(O) again at A′A'. Perpendicular bisector of OA′OA' cut BCBC at PAP_A. PB,PCP_B,P_C define similarly. Prove that :
I) Point PA,PB,PCP_A,P_B,P_C are collinear.
II ) Prove that the distance of OO from this line is equal to R2\frac {R}{2} where RR is the radius of the circumcircle.
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