SHARYGIN 2020 Problem 18

Bisectors $AA_1$ , $BB_1$ , and $CC_1$ of triangle $ABC$ meet at point $I$ . The perpendicular bisector to $BB_1$ meets $AA_1,CC_1$ at points $A_0,C_0$ respectively. Prove that the circumcircles of triangles $A_0IC_0$ and $ABC$ touch.