US-TST 2015 Problem 1

Let $A B C$ be a scalene triangle with incenter $I$ whose incircle is tangent to $\overline{B C}$, $\overline{C A}, \overline{A B}$ at $D, E, F$, respectively. Denote by $M$ the midpoint of $\overline{B C}$ and let $P$ be a point in the interior of $\triangle A B C$ so that $M D=M P$ and $\angle P A B=\angle P A C$. Let $Q$ be a point on the incircle such that $\angle A Q D=90^{\circ}$. Prove that either $\angle P Q E=90^{\circ}$ or $\angle P Q F=90^{\circ}$.