PMO 2008 Problem 3

Let $P$ be a point outside a circle $\Gamma$ , and let the two tangent lines through $P$ touch $\Gamma$ at $A$ and $B$ . Let $C$ be on the minor arc $AB$ , and let ray $PC$ intersect $\Gamma$ again at $D$ . Let $\ell$ be the line through $B$ and parallel to $PA$ . $\ell$ intersects $AC$ and $AD$ at $E$ and $F$ , respectively. Prove that $B$ is the midpoint of $EF$ .