Two circles of radius $6$ are externally tangent to each other and internally tangent to a circle of radius $15$ at points $A,B$. The distance $AB=\dfrac mn$. What is $m+n$?

1. Let $C,D$ be the centers of the two radius-$6$ circles. Since $O,C,A$ are collinear (internal tangency) and $O,D,B$ are collinear, $\triangle OCD\sim\triangle OAB$, with $CD=2\times6=12$ (the two small circles are externally tangent), $OC=15-6=9$, $OA=15$.
2. $AB=CD\cdot\dfrac{OA}{OC}=12\cdot\dfrac{15}9=20$.
3. $m+n=20+1=\boxed{21}$.