The Theorem: Given two non-intersecting circles, if a chain of n circles exists where each is tangent to both given circles and to its neighbors, and the chain closes (first tangent to last), then infinitely many such chains exist.
The Magic: You can start the chain at any point on the outer circle, and it will still close! This is proven using circle inversion to transform the problem into concentric circles where the result is obvious by symmetry.