KöMaL 2014 B4670

Let $A_1$, $B_1$ and $C_1$ be the midpoints of the sides of a triangle $ABC$ . Drop a perpendicular from $A_1$ to the angle bisector drawn from vertex $A$ , from $B_1$ to the angle bisector drawn from vertex $B$ , and from $C_1$ to the angle bisector drawn from vertex $C$ . Let $A_2$ denote the intersection of the perpendiculars from $B_1$ and from $C_1$. The points $B_2$ and $C_2$ are obtained in a similar way. Show that the lines $A_1A_2$, $B_1B_2$ and $C_1C_2$ are concurrent.