Question 151
Let $AB$ be a chord of circle $O$, $M$ the midpoint of arc $AB$, and $C$ a point outside of the circle $O$. From $C$ draw two tangents to the circle at points $S$, $T$. $MS \cap AB = E$, $MT \cap AB = F$. From $E$, $F$ draw a line perpendicular to $AB$, and intersecting $OS$, $OT$ at $X$, $Y$ respectively. Now draw a line from $C$ which intersects the circle $O$ at $P$ and $Q$. Let $R = MP \cap AB$ and let $Z$ be the circumcenter of $\triangle PQR$. Prove that $X$, $Y$, $Z$ are collinear.