Question 31
Let $ABC$ be an isosceles triangle with $AB = AC$, and let $M$ be the midpoint of side $BC$. Let $P$ be a point in the plane different from $A$ such that $PB \lt PC$ and $PA$ is parallel to $BC$. Let $X$ be a point on line $PB$ and $Y$ a point on line $PC$ such that $B$ lies on segment $PX$, $C$ lies on segment $PY$, and $\angle PXM = \angle PYM$. Prove that the four points $A, P, X, Y$ are concyclic.