Question 141
Let $D$ be a point on side $BC$ of triangle $ABC$ such that $\angle CAD = \angle CBA$. A circle with center $O$ passes through $B$, $D$, and meets segments $AB$, $AD$ at $E$, $F$, respectively. Lines $BF$ and $DE$ meet at point $G$. $M$ is the midpoint of $AG$. Prove that $CM \perp AO$. (Posed by Xiong Bin)