2013 OIM Problems/Problem 2

Problem

Let $X$ , $Y$ be the ends of a diameter of a circle $\Gamma$ and $N$ the the midpoint of one of the $XY$ arcs of $\Gamma$ . Let $A$ and $B$ be two points on the segment $XY$ . The straight lines $NA$ and $NB$ cut $\Gamma$ again at points $C$ and $D$ , respectively. The tangents to $\Gamma$ in $C$ and $D$ intersect at $P$ . Let $M$ be the point of intersection of segment $XY$ with segment $NP$ . Show that $M$ is the midpoint of segment $AB$ .