Apollonius problem in terms of oriented circles

TL;DR

This paper presents a novel approach to Apollonius' problem using oriented circles and inversive invariants, providing a comprehensive solution framework that accounts for tangency types and solution counts.

Contribution

It introduces a method based on oriented circles and inversive invariants to solve Apollonius' problem, including solutions for different tangency conditions.

Findings

Solution characterized by oriented circles and inversive invariants

Reversing circles yields additional solutions for the classical problem

Addresses tangency as tangent vector coincidence

Abstract

The solution of Apollonius' problem on constructing a circle (line), tangent to three given circles (lines), is presented in terms of oriented circles and inversive invariants. Tangency is understood as the coincidence of tangent vectors at the common point, in contrast to counter-tangency. The problem has 0, 1 or 2 solutions. By reversing each of the given circles one by one, we obtain the remaining solutions of the classical non-oriented problem.