A circle centered at $A$ with radius $4$ and a circle centered at $B$ with radius $9$ are externally tangent. A third circle is tangent to the first two and to one of their common external tangents. What is the radius of the third circle?

1. By theorem 48, $\frac1{\sqrt c}=\frac1{\sqrt a}+\frac1{\sqrt b}$.
2. With $a=4,b=9$: $\frac1{\sqrt c}=\frac12+\frac13=\frac56$, so $\sqrt c=\frac65$, $c=\boxed{\dfrac{36}{25}}$.