Question 24
$C_1$, $C_2$, $C_3$ are circles with radii $r_1$, $r_2$, $r_3$ respectively. The circles do not intersect and no circle lies inside any other circle. $C_1$ is larger than the other two. The two outer common tangents to $C_1$ and $C_2$ 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.