KöMaL 2006 B3946

The angle bisector drawn from vertex $C$ of an acute angled triangle ABC intersects the opposite side at the point $F$. The feet of the perpendiculars drawn from the point $F$ to the sides $BC$ and $CA$ are $P$ and $Q$, respectively. Let $M$ denote the intersection of the lines $AP$ and $BQ$. Prove that $AB$ and $CM$ are perpendicular to each other.