KöMaL 2010 A514

There are given three circles in the plane, $k_0$, $k_1$ and $k_2$, being externally tangent to each other. The center of $k_0$ is $O$ and one of its diameters is $A_1 A_2$. Denote by $B$the point of tangency between $k_1$ and $k_2$, by $C_1$ the point of tangency between $k_0$ and $k_1$ and by $C_2$ the point of tangency between $k_0$ and $k_2$. The line segments $A_1 C_2$ and $A_2 C_1$ meet at point $D$ in the interior of the circle $k_0$. Let $t_1$ and $t_2$ be the tangent lines to the circle $k_0$ at $A_1$ and $A_2$, respectively. Prove that if $t_1$ is tangent to $k_1$ and $t_2$ is tangent to $k_2$ then the line segment $OB$ passes through point $D$.