An even more straightforward proof of Descartes's circle theorem

TL;DR

This paper presents a simpler proof of Descartes's circle theorem using Cayley-Menger determinants, addressing why this approach was overlooked for nearly two centuries.

Contribution

It introduces a novel, straightforward proof of Descartes's circle theorem leveraging Cayley-Menger determinants, simplifying previous complex proofs.

Findings

Proof is significantly simpler than previous methods

Addresses historical oversight of this proof approach

Potentially broadens understanding of circle theorems

Abstract

How was this proof overlooked for 181 years? We give a simple proof of Descartes's circle theorem using Cayley-Menger determinants.