Question 22
In triangle $ABC$, with $AB \lt AC$, $I$ is the incenter, $E$ is the intersection of the $A$-excircle and $BC$. Point $F$ lies on the external angle bisector of $\angle BAC$ such that $E$ and $F$ lie on the same side of the line $AI$ and $\angle AIF = \angle AEB$. Point $Q$ lies on $BC$ such that $\angle AIQ = 90^\circ$. Circle $\omega_b$ is tangent to $FQ$ and to $AB$ at $B$, circle $\omega_c$ is tangent to $FQ$ and to $AC$ at $C$, and both circles pass through the inside of triangle $ABC$. If $M$ is the midpoint of the arc $BC$ which does not contain $A$, prove that $M$ lies on the radical axis of $\omega_b$ and $\omega_c$.
