SHARYGIN 2008 Problem 12

(A.Myakishev, 9--10) Given a triangle $ABC$ . Point $A_1$ is chosen on the ray $BA$ so that segments $BA_1$ and $BC$ are equal. Point $A_2$ is chosen on the ray $CA$ so that segments $CA_2$ and $BC$ are equal. Points $B_1$ , $B_2$ and $C_1$ , $C_2$ are chosen similarly. Prove that lines $A_1A_2$ , $B_1B_2$ , $C_1C_2$ are parallel.