In the following figure, two circles $A$ and $B$ tangent to each other and also tangent to line $CD$ at $D$ and $C$ respectively. If circle $A$ has radius $a$ and circle $B$ has radius $b$, then $CD=2\sqrt{ab}$.

Draw $AE\bot BC$, then $BE=b-a, AB=b+a$, so $CD^2=AE^2=AB^2-BE^2=(a+b)^2-(b-a)^2=4ab$, so $CD=2\sqrt{ab}$.