KöMaL 2021 B5203

In a triangle $ABC$, $AB>BC$, the inscribed circle touches sides $BC$, $CA$ and $AB$ at the points $A_0$, $B_0$ és $C_0$, respectively, and the escribed circle drawn to side $AC$ touches $AC$ at point $B_1$. Show that the intersection of the line segments $A_0B_1$ and $B_0C_0$ lies on the interior angle bisector drawn from vertex $B$ if and only if the angle at vertex $C$ measures $90^{\circ}$.