SHARYGIN 2023 Problem 82

The bisectors of angles $A$ , $B$ , and $C$ of triangle $ABC$ meet for the second time its circumcircle at points $A_1$ , $B_1$ , $C_1$ respectively. Let $A_2$ , $B_2$ , $C_2$ be the midpoints of segments $AA_1$ , $BB_1$ , $CC_1$ respectively. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.