The Six Circles Theorem revisited

TL;DR

This paper revisits the Six Circles Theorem, demonstrating that circle chains inscribed in a triangle are generally eventually 6-periodic, but can have arbitrarily long initial segments before repeating.

Contribution

It extends the classical theorem by showing that the chain's periodicity can have an arbitrarily long pre-period before settling into a 6-cycle.

Findings

Chains are eventually 6-periodic

Pre-period length can be arbitrarily long

The theorem's generalization broadens understanding of circle chains

Abstract

The Six Circles Theorem of C. Evelyn, G. Money-Coutts, and J. Tyrrell concerns chains of circles inscribed into a triangle: the first circle is inscribed in the first angle, the second circle is inscribed in the second angle and tangent to the first circle, the third circle is inscribed in the third angle and tangent to the second circle, and so on, cyclically. The theorem asserts that if all the circles touch the sides of the triangle, and not their extensions, then the chain is 6-periodic. We show that, in general, the chain is eventually 6-periodic but may have an arbitrarily long pre-period.