KZ-MO 2000 Problem 2

Given a circle centered at $O$ and two points $A$ and $B$ lying on it. $A$ and $B$ do not form a diameter. The point $C$ is chosen on the circle so that the line $AC$ divides the segment $OB$ in half. Let lines $AB$ and $OC$ intersect at $D$ , and let lines $BC$ and $AO$ intersect at $F$ . Prove that $AF = CD$ .