Tangent circles between two parent circles
A Steiner chain is a sequence of circles, each tangent to two fixed "parent" circles and to its neighbors in the chain.
Steiner's Porism: If a closed chain of n circles exists for two parent circles, then infinitely many closed n-chains exist—you can start the chain anywhere!
This is proven by inversion: the configuration maps to concentric circles where symmetry makes the result obvious.