Inner and outer Soddy circles (red).
A Soddy circle is a solution to the three-circle form of the Apollonius problem in which each of the given circles is tangent to (just touches) the other two.1 There are two Soddy circles: the outer Soddy circle, which surrounds the three given circles, and the inner Soddy circle, which is interior to them. The inner Soddy circle is the solution to the four coins problem.
1. Soddy, F. "The Kiss Precise." Nature, 137: 1021, 1936.