Question 15
Let $ABC$ be an isosceles triangle with $AB = AC$. Suppose $P, Q, R$ are points on segments $AC, AB, BC$ respectively such that $AP = QB$, $\angle PBC = 90 - \angle BAC$ and $RP = RQ$. Let $O_1, O_2$ be the circumcenters of $\triangle APQ$ and $\triangle CRP$. Prove that $BR = O_1O_2$.