Generalized problem of Apollonius

TL;DR

This paper generalizes Apollonius' problem by analyzing the maximum number of tangent solution circles for more than three given circles, establishing upper bounds and characterizing special cases.

Contribution

It extends Apollonius' problem to more than three circles, providing bounds on the number of solutions and characterizing quadruples with maximum solutions.

Findings

At most six solutions for four circles

At most four solutions for five circles

Characterization of quadruples with six solutions

Abstract

The aim of this paper is to generalize Apollonius' problem. The problem is to construct a circle that is tangent to three given circles in a plane. We find the maximum possible number of solution circles in the case of more than the three given circles. We show that if all the given circles are not tangent at the same point, then there exist at most six solutions in the case of the four given generalized circles and there exist at most four solutions in the case of the five given generalized circles. We also describe all quadruples of generalized circles with exactly six solutions.