A theorem on circle configurations

TL;DR

This paper presents a general theorem providing formulas for the radii and positions of four circles in arbitrary configurations, extending to n-spheres, and includes special cases like the Descartes Theorem and Apollonian problem solutions.

Contribution

It introduces a new general theorem that analytically describes circle configurations and extends to higher-dimensional spheres, unifying several classical results.

Findings

Derived formulas for four-circle configurations

Extended Descartes Theorem included

Analytic solutions to the Apollonian problem

Abstract

A formula for the radii and positions of four circles in the plane for an arbitrary linearly independent circle configuration is found. Among special cases is the recent extended Descartes Theorem on the Descartes configuration and an analytic solution to the Apollonian problem. The general theorem for n-spheres is also considered.