Question 168
Let $k$ be the circumcircle of a given convex quadrilateral $ABCD$ with the property that the half-lines $DA$ and $CB$ meet at a point $E$ for which $|CD|^2 = |AD| \cdot |ED|$ holds. Let us denote by $F$ ($F \neq A$) the point of intersection of the circle $k$ with the perpendicular to $ED$ at $A$. Prove that the segments $AD$ and $CF$ are congruent if and only if the circumcenter of the triangle $ABE$ lies on $ED$.