Question 140
Let $ABC$ be a scalene triangle. Let $A_1$, $B_1$, $C_1$ be the points on the sides $BC$, $CA$, $AB$, respectively such that $\angle B_1AC_1 = \angle B_1A_1C_1$, $\angle C_1BA_1 = \angle C_1B_1A_1$ and $\angle A_1CB_1 = \angle A_1C_1B_1$. Let $A_2$, $B_2$, $C_2$ be the second intersections of the circumcircle of $A_1B_1C_1$ with the segments $BC$, $CA$, $AB$, respectively.
Prove that the lines $AA_2$, $BB_2$, $CC_2$ are concurrent if and only if $\angle BC_1A_1 = \angle A_1B_1C$.
Prove that the lines $AA_2$, $BB_2$, $CC_2$ are concurrent if and only if $\angle BC_1A_1 = \angle A_1B_1C$.