TECHNOLOGY ILLUSTRATIONS OF FIVE SOLUTIONS
TOTHE PROBLEM OF APOLLONIUS
Michael I. Ratliff and Janet McShane
Department of Mathematics and Statistics
Box 5717
Northern Arizona University
Flagstaff, AZ 860111-5717
928-523-6895 (Ratliff) Michael.Ratliff@nau.edu
928-523-1252 (McShane) Janet.McShane@nau.edu
Eves (1964) calls Euclid, Archimedes and Apollonius "the three
mathematical giants of the third century B.C.". 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. The problem calls for "constructing a circle tangent to
three given circles, where the given circles are permitted to
degenerate independently into straight lines or points". Several
mathematicians were attracted to this problem including Vieta,
Newton, and Descartes. Gow (1968) describes how these mathematicians
solved the problem. For instance, Vieta (1540-1603) proposed the
problem to Romanus, who solved it by finding that the center of
the tangent circle is the intersection of two parabolas. Vieta
instead solved it very elegantly using ordinary geometry, while
Newton reduced the intersection of the two loci, which Romanus
had constructed, to the intersection of two straight lines.
Descartes on his part approached the problem algebraically and
Gow (1968) comments that "of the two solutions which he found,
he admits that one gave an expression so complicated that he
would not undertake to construct it in a month".
Nowadays several solutions to the Problem of Apollonius can be
investigated and illustrated with the aid of technology, which
allows us to easily and almost instantaneously make constructions,
and to manipulate symbolic expressions. 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. Since all the methods involve the use of
technology, the most desirable tools to have access to are a
symbolic manipulator and dynamic software. We will use the
symbolic mathematical software Mathematica and the dynamic
geometric software Geometer's Sketchpad.