Question 160
Let $M$ be the midpoint of the side $BC$ of an acute $\triangle ABC$, and let $D$ be the foot of perpendicular from $C$ to $AM$. The circumcircle of $\triangle ABD$ intersects the side $BC$ again at $E \neq B$. Suppose $F$ is a point on the segment $AE$ such that $FB = FC$. Prove that $F$ is the midpoint of $AE$.