Question 71
Points $A_1$ and $A_2$ are the intersection points of the incircle and the angle bisector of vertex $A$ with side $BC$ of triangle $ABC$, respectively. Points $B_1$, $B_2$, $C_1$ and $C_2$ are defined similarly. Suppose that the perpendicular from $A_1$ to line $B_2C_2$ intersects the angle bisector of vertex $A$ in $A'$. Points $B'$ and $C'$ are defined similarly. Prove that the two triangles $A_1B_1C_1$ and $A'B'C'$ are congruent.
