Question 106
Let $AB$ be a chord on a circle $O$, and let $M$ be the midpoint of the smaller arc $AB$. From a point $C$ outside the circle $O$ draw the two tangents to the circle $O$, touching it at $S$ and $T$. Suppose $MS$ meets $AB$ at the point $E$, and $MT$ meets $AB$ at the point $F$. From $E$ and $F$ draw the lines perpendicular to $AB$, meeting $OS$ and $OT$ at $X$ and $Y$, respectively. Draw another line from $C$ meeting the circle $O$ at the points $P$ and $Q$. Let $R$ be the intersection point of $MP$ and $AB$. Finally, let $Z$ be the circumcenter of $\triangle PQR$.
Prove that $X$, $Y$, $Z$ are collinear.
Prove that $X$, $Y$, $Z$ are collinear.