Closing Theorems for Circle Chains

Year 2025, Volume: 18 Issue: 2, 259 - 276, 19.10.2025

Norbert Hungerbühler

https://doi.org/10.36890/iejg.1624214

Abstract

We consider closed chains of circles $C_1,C_2,\ldots,C_n,C_{n+1}=C_1$ such that two neighbouring circles $C_i,C_{i+1}$ intersect or touch each other with $A_i$ being a common point. We formulate conditions such that a polygon with vertices $X_i$ on $C_i$, and $A_i$ on the (extended) side $X_iX_{i+1}$, is closed for every position of the starting point $X_1$ on $C_1$. Similar results apply to open chains of circles. It turns out that the intersection of the sides $X_iX_{i+1}$ and $X_jX_{j+1}$ of the polygon lies on a circle $C_{ij}$ through $A_i$ and $A_j$ with the property that $C_{ij}, C_{jk}$ and $C_{ki}$ pass through a common point. The six circles theorem of Miquel and Steiner's quadrilateral Theorem appear as special cases of the general results.

Keywords

Closing theorems , circle chains , Miquel’s six circles theorem , Steiner’s quadrilateral theorem

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