Middle European Mathematical Olympiad 2013 Problem T-6

Let $K$ be a point inside an acute triangle $ABC$, such that $BC$ is a common tangent of the circumcircles of $AKB$ and $AKC$. Let $D$ be the intersection of the lines $CK$ and $AB$, and let $E$ be the intersection of the lines $BK$ and $AC$. Let $F$ be the intersection of the line $BC$ and the perpendicular bisector of the segment $DE$. The circumcircle of $ABC$ and the circle $k$ with centre $F$ and radius $FD$ intersect at points $P$ and $Q$.

Prove that the segment $PQ$ is a diameter of $k$.