A Perpendicularity from Two Isosceles Triangles

An acute triangle $ABC$ is inscribed in a circle $O$. The perpendicular from $A$ to $BC$ meets the circle $O$ again at $D$. A point $K$ lies on segment $BC$, and points $E,F$ lie on $AC,AB$ with $FB=FK$ and $EC=EK$. Prove that $DK\perp EF$.