RU-ALLRUSSIAN 1984 Problem 23

$\mathbf{C}_{1}$ , $\mathbf{C}_{2}$ , $\mathbf{C}_{3}$ are circles with radii $\mathbf{r}_{1}$ , $\mathbf{r}_{2}$ , $\mathbf{r}_{3}$ respectively. The circles do not intersect and no circle lies inside any other circle. $\mathbf{C}_{1}$ is larger than the other two. The two outer common tangents to $\mathbf{C}_{1}$ and $\mathbf{C}_{2}$ meet at A ("outer" means that the points where the tangent touches the two circles

lie on the same side of the line of centers). The two outer common tangents to $C_1$ and $C_3$ intersect at B. The two tangents from A to $C_3$ and the two tangents from B to $C_2$ form a quadrangle. Show that it has an inscribed circle and find its radius.