KöMaL 2011 B4396

The inscribed circle of triangle ABC touches side $b$ at $F$ and side $a$ at $G$. Prove that the foot of the perpendicular dropped from $A$ onto the angle bisector of angle $\beta$ lies on the line $FG$. (By conventional notation , side $b$ lies opposite to vertex $B$ and angle $\beta$ is the angle at $B$, etc.)