SHARYGIN 2016 Problem 14

Let a triangle $ABC$ be given. Consider the circle touching its circumcircle at $A$ and touching externally its incircle at some point $A_1$ . Points $B_1$ and $C_1$ are defined similarly. a) Prove that lines $AA_1, BB_1$ and $CC1$ concur. b) Let $A_2$ be the touching point of the incircle with $BC$ . Prove that lines $AA_1$ and $AA_2$ are symmetric about the bisector of angle $\angle A$ .