Question 74
Let $\mathcal{C}$, $\mathcal{C}_1$, $\mathcal{C}_2$ be three circumferences of radii respectively $r$, $r_1$, $r_2$ with $0 \lt r_1 \lt r_2 \lt r$. The circles $\mathcal{C}_1$ and $\mathcal{C}_2$ are internally tangent to $\mathcal{C}$ at two distinct points $A$ and $B$ and intersect each other at two distinct points. Prove that the segment $AB$ passes through one of the intersection points between $\mathcal{C}_1$ and $\mathcal{C}_2$ if and only if $r_1 + r_2 = r$.