Question 127
For acute triangle $ABC$ with $AB \gt AC$, let $M$ be the midpoint of side $BC$ and $P$ a point inside $\triangle AMC$ such that $\angle MAB = \angle PAC$. Let $O$, $O_1$ and $O_2$ be the circumcenters of $\triangle ABC$, $\triangle ABP$ and $\triangle ACP$ respectively. Prove that line $AO$ bisects segment $O_1O_2$. (Posed by Xiong Bin)