IT-CESENATICO 1999 Problem 3

Let $r_1,r_2,r$ , with $r_1 < r_2 < r$ , be the radii of three circles $\Gamma_1,\Gamma_2,\Gamma$ , respectively. The circles $\Gamma_1,\Gamma_2$ are internally tangent to $\Gamma$ at two distinct points $A,B$ and intersect in two distinct points. Prove that the segment $AB$ contains an intersection point of $\Gamma_1$ and $\Gamma_2$ if and only if $r_1 +r_2 = r$ .