Question 144
Suppose points $O$ and $I$ are the circumcenter and incenter of $\triangle ABC$ respectively, and the inscribed circle of $\triangle ABC$ is tangent to the sides $BC$, $CA$, $AB$ at points $D$, $E$, $F$ respectively. Lines $FD$ and $CA$ intercept at point $P$, while lines $DE$ and $AB$ intercept at point $Q$. And points $M$, $N$ are the midpoint of segments $PE$, $QF$ respectively. Prove that $OI \perp MN$.