KöMaL 2009 B4179

The vertex $C$ of a parabola is the centre of a circle that passes through the focus $F$ of the parabola. Let the intersections of the parabola and the circle be $A$ and $B$, let $AB$and $CF$ intersect at $E$, and let $D$ denote the point of the circle that lies opposite to $F$. Show that $DE$ is the geometric mean of $FE$ and the diameter of the circle.