BALTIC-WAY 2009 Problem 11

Let $M$ be the midpoint of the side $AC$ of a triangle $ABC$ , and let $K$ be a point on the ray $BA$ beyond $A$ . The line $KM$ intersects the side $BC$ at the point $L$ . $P$ is the point on the segment $BM$ such that $PM$ is the bisector of the angle $LPK$ . The line $\ell$ passes through $A$ and is parallel to $BM$ . Prove that the projection of the point $M$ onto the line $\ell$ belongs to the line $PK$ .