IR-MO 1998 Problem 5

In a triangle $ABC$ , the bisector of angle $BAC$ intersects $BC$ at $D$ . The circle $\Gamma$ through $A$ which is tangent to $BC$ at $D$ meets $AC$ again at $M$ . Line $BM$ meets $\Gamma$ again at $P$ . Prove that line $AP$ is a median of $\triangle ABD.$