Technology Illustrations of Five Solutions to the
Problem of Apollonius

Michael Ratliff and Janet McShane
Northern Arizona University

Eves (1964) call Euclid, Archimedes and Apollonius “the three mathematical giants of the third century B.C.,” (p.149).  Apollonius’ (ca. 260-170 B.C.) fame comes from his extensive work on conic sections.  He also provided us with one of the most famous classical construction problems.  This problem is now known as the Problem of Apollonius and calls for “constructing a circle tangent to three given circles, where the three given circles are permitted to degenerate independently into straight lines or points.” (Eves, 1964, p. 152).  Several mathematicians were attracted to this problem including Vieta, Newton, and Descartes.  Some approached the problem geometrically while others approached the problem algebraically. 

Nowadays several solutions to The Apollonius Problem can be investigated and illustrated with the aid of technology, which allow us to easily and almost instantaneously make constructions, and to manipulate symbolic expressions.  Today we discuss five methods of solution to this famous problem, four of which use dynamic software and one that uses analytic geometry.  Each method initially discusses the problem when three (non-overlapping) circles are given, but reference is made to cases having overlapping circles, points, and lines.