Let $ABC$ be an acute triangle with $AB<AC$. $E,F$ are foots of the altitudes drawn from $B,C$ respectively. Let $M$ be the midpoint of segment $BC$. The tangent at $A$ to the circumcircle of $ABC$ cuts $BC$ in $P$ and $EF$ cuts the parallel to $BC$ from $A$ at $Q$. Prove that $PQ$ is perpendicular to $AM$.