Altitudes, an angle bisector and the midpoint of BC
Problem
Let $ABC$ be a triangle with $\angle A \neq 90^\circ$, and let $BE$ and $CF$ be its altitudes. The bisector of angle $A$ meets $EF$ at $M$ and $BC$ at $N$. Let $P$ be the point with $MP \perp EF$ and $NP \perp BC$. Prove that the line $AP$ passes through the midpoint of $BC$.
The diagram
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