CWMO 2005 Problem 5

Circles $C(O_1)$ and $C(O_2)$ intersect at points $A$ , $B$ . $CD$ passing through point $O_1$ intersects $C(O_1)$ at point $D$ and tangents $C(O_2)$ at point $C$ . $AC$ tangents $C(O_1)$ at $A$ . Draw $AE \bot CD$ , and $AE$ intersects $C(O_1)$ at $E$ . Draw $AF \bot DE$ , and $AF$ intersects $DE$ at $F$ . Prove that $BD$ bisects $AF$ .