ÖMO 2016 Problem 2

Let $ABC$ be a triangle. Its incircle meets the sides $BC, CA$ and $AB$ in the points $D, E$ and $F$ , respectively. Let $P$ denote the intersection point of $ED$ and the line perpendicular to $EF$ and passing through $F$ , and similarly let $Q$ denote the intersection point of $EF$ and the line perpendicular to $ED$ and passing through $D$ . Prove that $B$ is the mid-point of the segment $PQ$ .

Proposed by Karl Czakler