OMCC 2012 Problem 2

Let $\gamma$ be the circumcircle of the acute triangle $ABC$ . Let $P$ be the midpoint of the minor arc $BC$ . The parallel to $AB$ through $P$ cuts $BC, AC$ and $\gamma$ at points $R,S$ and $T$ , respectively. Let $K \equiv AP \cap BT$ and $L \equiv BS \cap AR$ . Show that $KL$ passes through the midpoint of $AB$ if and only if $CS = PR$ .