Circle tangent to two other circles How can i find a circle that is tangent to two circles which have the same center?  Specifically i'm looking for a circle that will contain the smaller circle. I know how to find the circle whose diameter is equal to the difference between the two radii.
This picture may help.
 A: 
If the radius of the small circle is $|\overline{OP}| = p$ and the radius of the big circle is $|\overline{OQ}| = q$, then the diameter of the target circle is $p+q$, so that its radius is $\frac12(p+q)$.
Note that the points of tangency $P$ and $Q$, along with $O$ (the common center of the original circles) and $M$ (the center of the target circle), are collinear.
So, to "find" the target circle given point $P$, simply draw $\overrightarrow{PO}$; the intersection with the other circle is $Q$, and the midpoint of $\overline{PQ}$ is $M$. (Or, starting with $Q$, draw $\overrightarrow{QO}$; etc.) In terms of vectors or coordinates (with $O$ not-necessarily the origin), you can write
$$p\;( Q - O ) = -q\;( P - O )$$
so that
$$M = \frac12\left(\; P + Q \;\right) = \frac{1}{2p} \left(\; (p+q) O + (p-q) P \;\right) = \frac{1}{2q}\left(\; (p+q) O + ( q - p ) Q \;\right)$$
A: Take a look at www.arcenciel.co.uk/geometry/index.htm 
They have proofs and C# coding for many types of circle tangentiality
