CMO 1990 Problem 1

Given a convex quadrilateral $ABCD$ , side $AB$ is not parallel to side $CD$ . The circle $O_1$ passing through $A$ and $B$ is tangent to side $CD$ at $P$ . The circle $O_2$ passing through $C$ and $D$ is tangent to side $AB$ at $Q$ . Circle $O_1$ and circle $O_2$ meet at $E$ and $F$ . Prove that $EF$ bisects segment $PQ$ if and only if $BC\parallel AD$ .