Problem
GEO-B3-M01-P026 Porism Between Two Circles
Two disjoint circles \(R_1\) and \(R_2\) have a closed chain of \(n\) circles tangent to both given circles and to their neighbors in the chain. Prove that the initial circle may be chosen arbitrarily among circles tangent to \(R_1\) and \(R_2\) in the same way: after \(n\) steps the chain will close again.
C. Hint 1. Transform the given circles into concentric circles.
D. Hint 2. In the concentric model, one step of the chain is a rotation by a fixed angle.
E. Full solution. There is an inversion, possibly followed by a homothety, sending \(R_1\) and \(R_2\) to concentric circles. Tangencies are preserved. Between two concentric circles, all circles tangent to both are equal; their centers lie on a circle concentric with the given ones. If one closed chain of \(n\) circles exists, then passing from the center of one chain circle to the next is a rotation by a fixed angle \(\varphi\), with \(n\varphi=360^\circ\). Hence, starting from any other position of the first circle, we repeat the same rotation; after \(n\) steps we make a full turn and return to the beginning. The inverse transformation gives the required chain for the original circles.
This is final-level: the main step is not computation, but recognizing a model where the whole configuration is controlled by rotation.