Let $k$ be a circle with diameter $AB$ and centre $O$. Let C be an arbitrary point on the circle different from $A$ and $B$. Let $D$ be the point for which $O$, $B$, $D$ and $C$ (in this order) are the four vertices of a parallelogram. Let $E$ be the intersection of the line $BD$ and the circle $k$, and let $F$ be the orthocenter of the triangle $OAC$. Prove that the points $O, D, E, C, F$ lie on a circle.