HMMT 2012 S9

Let $A B C$ be a triangle with $A B<A C$. Let $M$ be the midpoint of $B C$. Line $l$ is drawn through $M$ so that it is perpendicular to $A M$, and intersects line $A B$ at point $X$ and line $A C$ at point $Y$. Prove that $\angle B A C=90^{\circ}$ if and only if quadrilateral $X B Y C$ is cyclic.