Conways Circle Theorem: A Short Proof Enabling Generalization to Polygons

TL;DR

This paper presents a concise proof of Conway's Circle Theorem using tangent properties of the incircle, which also allows for generalization from triangles to polygons in plane geometry.

Contribution

A novel, simplified proof of Conway's Circle Theorem that introduces a perspective enabling its extension to polygons.

Findings

The proof simplifies understanding of the theorem.

The approach generalizes the theorem to polygons.

It highlights the role of incircle tangents in geometric proofs.

Abstract

John Conway's Circle Theorem is a gem of plane geometry. The six points formed by continuing the sides of a triangle beyond every vertex by the length of its opposite side, are concyclic. The theorem has attracted several proofs. We present a short proof that views the extended sides as equal tangents of the incircle, a perspective that enables generalization to polygons.