KöMaL 2011 B4398

$A_1$, $B_1$ and $C_1$ are given points on sides $BC$, $CA$ and $AB$, respectively, of the acute-angled triangle ABC . Mark a point $A_2$ on the circle $AC_1 B_1$. Let $B_2$ be the intersection (different from $C_1$) of the line $C_1 A_2$ with the circle $BA_1 C_1$. Furthermore, let $C_2$ be the intersection (different from $A_1$) of the line $A_1 B_2$ and the circle $CB_1 A_1$. Prove that the points $A_2$, $B_1$ and $C_2$ are collinear.