Apollonius’ Problem: A Study of Solutions and Their Connections

Abstract

In Tangencies Apollonius of Perga showed how to construct a circle that is tangent to three given circles. More generally, Apollonius' problem asks to construct the circle which is tangent to any three objects that may be any combination of points, lines, and circles. The case when all three objects are circles is the most complicated case since up to eight solution circles are possible depending on the arrangement of the given circles. Within the last two centuries, solutions have been given by J. D. Gergonne in 1816, by Frederick Soddy in 1936, and most recently by David Eppstein in 2001. In this report, we illustrate the solution using the geometry software Cinderella™, survey some connections among the three solutions, and provide a framework for further study.