KöMaL 2012 A564

Let $k$ be the incircle in the triangle ABC , which is tangent to the sides $AB$, $BC$, $CA$ at the points $C_0$, $A_0$ and $B_0$, respectively. The angle bisector starting at $A$ meets $k$ at $A_1$ and $A_2$, the angle bisector starting at $B$meets $k$ at $B_1$ and $B_2$; $AA_1$<$AA_2$ and $BB_1$<$BB_2$. The circle $k_1 \ne k$ is tangent externally to the side $CA$ at $B_0$ and it is tangent to the line $AB$. The circle $k_2 \ne k$ is tangent externally to the side $BC$ at $A_0$, and it is tangent to the line $AB$. The circle $k_3$ is tangent to $k$ at $A_1$, and it is tangent to $k_1$ at point $P$. The circle $k_4$ is tangent to $k$ at $B_1$, and it is tangent to $k_2$ at point $Q$. Prove that the radical axis between the circles $A_1 A_2 P$ and $B_1 B_2 Q$ is the angle bisector starting at $C$.