KMO 2005 Problem 5

Let $P$ be a point that lies outside of circle $O$ . A line passes through $P$ and meets the circle at $A$ and $B$ , and another line passes through $P$ and meets the circle at $C$ and $D$ . The point $A$ is between $P$ and $B$ , $C$ is between $P$ and $D$ . Let the intersection of segment $AD$ and $BC$ be $L$ and construct $E$ on ray $(PA$ so that $BL \cdot PE = DL \cdot PD$ .

Show that $M$ is the midpoint of the segment $DE$ , where $M$ is the intersection of lines $PL$ and $DE$ .