Question 32
Let $ABC$ be an acute triangle with circumcenter $O$ and incenter $I$. Points $E, M$ lie on $AC$ and $F, N$ on $AB$ so that $BE \perp AC$, $CF \perp AB$, $\angle ABM = \angle CBM$ and $\angle ACN = \angle BCN$. Prove that $I$ lies on $EF$ if and only if $O$ lies on $MN$.