Question 111
In the plane, let $O$ be the centre of the circle $\Gamma$, and let $A$, $B$ be two points on $\Gamma$ such that $O, A, B$ are not collinear. Let $M$ be the midpoint of segment $AB$, and take points $P$, $Q$ on lines $OA$, $OB$ respectively such that $P \neq A$ and $P, M, Q$ are collinear. Let the line through $P$ parallel to $AB$ and the line through $Q$ parallel to $OM$ meet at $X$; and let the line through $X$ parallel to $OA$ and the line through $B$ perpendicular to $OX$ meet at $Y$. Prove that if $X$ lies on $\Gamma$, then $Y$ also lies on $\Gamma$.