MEMO 2013 Problem 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 $k$ .