DE-MO 2006 Problem 6

Let a circle through $B$ and $C$ of a triangle $ABC$ intersect $AB$ and $AC$ in $Y$ and $Z$ , respectively. Let $P$ be the intersection of $BZ$ and $CY$ , and let $X$ be the intersection of $AP$ and $BC$ . Let $M$ be the point that is distinct from $X$ and on the intersection of the circumcircle of the triangle $XYZ$ with $BC$ . Prove that $M$ is the midpoint of $BC$