USAMO 2016 — Days 1 and 2 Question 3

Let $ABC$ be an acute triangle and let $I_B$, $I_C$, and $O$ denote its $B$-excenter, $C$-excenter, and circumcenter, respectively. Points $E$ and $Y$ are selected on $\overline{AC}$ such that $\angle ABY = \angle CBY$ and $\overline{BE} \perp \overline{AC}$. Similarly, points $F$ and $Z$ are selected on $\overline{AB}$ such that $\angle ACZ = \angle BCZ$ and $\overline{CF} \perp \overline{AB}$. Lines $I_B F$ and $I_C E$ meet at $P$. Prove that $\overline{PO}$ and $\overline{YZ}$ are perpendicular.

Proposed by Evan Chen, Telv Cohl (credited by Evan Chen).