IR-TST 2018 Problem 4

Let $ABC$ be a triangle ( $\angle A\neq 90^\circ$ ). $BE,CF$ are the altitudes of the triangle. The bisector of $\angle A$ intersects $EF,BC$ at $M,N$ . Let $P$ be a point such that $MP\perp EF$ and $NP\perp BC$ . Prove that $AP$ passes through the midpoint of $BC$ .

Proposed by Iman Maghsoudi, Hooman Fattahi