CAPS 2007 Problem 3

A convex quadrilateral $ABCD$ inscribed in a circle $k$ has the property that the rays $DA$ and $CB$ meet at a point $E$ for which $CD^2=AD\cdot ED.$ The perpendicular to $ED$ at $A$ intersects $k$ again at point $F.$ Prove that the segments $AD$ and $CF$ are congruent if and only if the circumcenter of $\triangle ABE$ lies on $ED.$