SHARYGIN 2018 Problem 17

Let each of circles $\alpha, \beta, \gamma$ touches two remaining circles externally, and all of them touche a circle $\Omega$ internally at points $A_1, B_1, C_1$ respectively. The common internal tangent to $\alpha$ and $\beta$ meets the arc $A_1B_1$ not containing $C_1$ at point $C_2$ . Points $A_2$ , $B_2$ are defined similarly. Prove that the lines $A_1A_2, B_1B_2, C_1C_2$ concur.