SHARYGIN 2013 Problem 10

The incircle of triangle $ABC$ touches the side $AB$ at point $C'$ ; the incircle of triangle $ACC'$ touches the sides $AB$ and $AC$ at points $C_1, B_1$ ; the incircle of triangle $BCC'$ touches the sides $AB$ and $BC$ at points $C_2$ , $A_2$ . Prove that the lines $B_1C_1$ , $A_2C_2$ , and $CC'$ concur.