Closing Theorems for Circle Chains

TL;DR

This paper establishes conditions for polygons inscribed in chains of circles to be closed, revealing geometric properties involving circle intersections and generalizing classical theorems like Miquel's and Steiner's.

Contribution

It introduces new closure conditions for polygons on circle chains and generalizes classical circle theorems within this framework.

Findings

Conditions for polygon closure in circle chains

Existence of circles through intersection points of polygon sides

Generalization of Miquel and Steiner theorems

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.