$ABCD$ is a cyclic quadrilateral. Lines $AB,CD$ intersect at $E$, lines $AD,BC$ intersect at $F$, and $EM$ and $FN$ are tangents to the circumcircle of $ABCD$. Two circles are constructed with $E,F$ their centers and $EM, FN$ their radii, respectively. $K$ is one of their intersections. Prove that $EK$ is perpendicular to $FK$.