JBMO-SL 2003 Problem 5

Let $ABC$ be an isosceles triangle with $AB = AC$ . A semi-circle of diameter $[EF]$ with $E, F \in [BC]$ , is tangent to the sides $AB,AC$ in $M, N$ respectively and $AE$ intersects the semicircle at $P$ . Prove that $PF$ passes through the midpoint of $[MN]$ .