Problem
GEO-B3-M01-P024 Porism of a Chain
Two disjoint circles \(R_1\) and \(R_2\) admit a closed chain of \(n\) circles, each tangent to \(R_1\), \(R_2\), and its two neighboring circles in the chain. Prove that if the first circle is replaced by any other circle tangent to \(R_1\) and \(R_2\) in the same way, the chain can again be closed after \(n\) steps.
Hint 1. Send \(R_1,R_2\) to concentric circles.
Hint 2. In the concentric model all chain circles are equal.
E. Full solution. There is an inversion followed by a homothety sending \(R_1\) and \(R_2\) to concentric circles. Tangencies are preserved. In the annular region between two concentric circles, every circle tangent to both has the same radius, so all chain circles become equal. A closed chain of \(n\) equal circles between concentric circles means that moving from one circle to the next is a rotation by a fixed angle \(\varphi\), with \(n\varphi=360^\circ\). If we start with any other such circle, the same rotation constructs the next circle; after \(n\) steps the full rotation returns to the starting circle. Applying the inverse transformation proves the statement for the original circles.
This is the final problem of the base block: inversion does not merely simplify the diagram, but turns the statement into rotational symmetry.