Question 122
Let $ABC$ be a triangle. Let $A_B$ denote the foot of the perpendicular line from $A$ to the exterior angle bisector of $B$, and let $A_C$ denote the foot of the perpendicular line from $A$ to the exterior angle bisector of $C$. The points $B_A, B_C, C_A, C_B$ are defined similarly. Prove that the hexagon $A_B A_C B_A B_C C_A C_B$ is cyclic.