OIM 2013 Problem 2

Let $X$ and $Y$ be the diameter's extremes of a circunference $\Gamma$ and $N$ be the midpoint of one of the arcs $XY$ of $\Gamma$ . Let $A$ and $B$ be two points on the segment $XY$ . The lines $NA$ and $NB$ cuts $\Gamma$ again in $C$ and $D$ , respectively. The tangents to $\Gamma$ at $C$ and at $D$ meets in $P$ . Let $M$ the the intersection point between $XY$ and $NP$ . Prove that $M$ is the midpoint of the segment $AB$ .