Question 132
Let $O$ be the circumcircle of the triangle $ABC$. Two circles $O_1$, $O_2$ are tangent to each of the circle $O$ and the rays $\overrightarrow{AB}$, $\overrightarrow{AC}$, with $O_1$ interior to $O$, $O_2$ exterior to $O$. The common tangent of $O$, $O_1$ and the common tangent of $O$, $O_2$ intersect at the point $X$. Let $M$ be the midpoint of the arc $BC$ (not containing the point $A$) on the circle $O$, and the segment $\overline{AA'}$ be a diameter of $O$. Prove that $X$, $M$, and $A'$ are collinear.