Problem of Apollonius

Classic Problem
Let there be three circles in the plane.

It is required to draw another circle tangent to each of the three.

Solution
It is generically possible to construct such a circle in $8$ different ways.

Each tangent circle encloses a subset of the three original circles.

From Cardinality of Power Set of Finite Set, there are thus $2^3 = 8$ distinct such tangent circles