Here are three "mutually tangent" circles. Essentially everything in this diagram can be dragged.

Some things to think about:

If I choose three points that do not lie on the same line together, can they always be used as the centers of three mutually-tangent circles? If so, how do I construct those circles? In other words, how do I find their radii?

Given three mutually-tangent circles, how can we find the area between them (shown in blue above)?

If I have two mutually-tangent circles of different sizes, how can I construct a third circle tangent to both of them? Or, to put it another way, where must the center of the third circle be? What sort of curve must it lie along?