MathNet 2013 Question 75

Let $K$ be a point inside an acute triangle $A B C$, such that $B C$ is a common tangent of the circumcircles of $A K B$ and $A K C$. Let $D$ be the intersection of the lines $C K$ and $A B$, and let $E$ be the intersection of the lines $B K$ and $A C$. Let $F$ be the intersection of the line $B C$ and the perpendicular bisector of the segment $D E$. The circumcircle of $A B C$ and the circle $k$ with centre $F$ and radius $F D$ intersect at points $P$ and $Q$.
Prove that the segment $P Q$ is a diameter of $k$.