Given three objects, each of which may be a Point, Line, or Circle, draw a Circle that is
Tangent to each. There are a total of ten cases. The two easiest involve three points or three Lines,
and the hardest involves three Circles. Euclid solved the two easiest cases in his *Elements*, and the others (with the exception of the three Circle problem), appeared in the *Tangencies* of
Apollonius which was, however, lost. The general problem is, in principle, solvable by Straightedge and Compass
alone.

The three-Circle problem was solved by Viète (Boyer 1968), and the solutions are called Apollonius Circles.
There are eight total solutions. The simplest solution is obtained by solving the three simultaneous quadratic equations

(1) |

(2) |

(3) |

(4) |

(5) |

(6) |

(7) | |||

(8) | |||

(9) | |||

(10) |

and similarly for , , and (where the 2 subscripts are replaced by 3s). Solving these two simultaneous linear equations gives

(11) | |||

(12) |

which can then be plugged back into the Quadratic Equation (1) and solved using the Quadratic Formula.

Perhaps the most elegant solution is due to Gergonne. It proceeds by locating the six Homothetic Centers (three internal and three external) of the three given Circles. These lie three by three on four
lines (illustrated above). Determine the Poles of one of these with respect to each of the three
Circles and connect the Poles with the Radical Center of the
Circles. If the connectors meet, then the three pairs of intersections are the points of tangency of two of
the eight circles (Johnson 1929, Dörrie 1965). To determine *which* two of the eight Apollonius circles are produced by
the three pairs, simply take the two which intersect the original three Circles only in a single point of
tangency. The procedure, when repeated, gives the other three pairs of Circles.

If the three Circles are mutually tangent, then the eight solutions collapse to two, known as the Soddy Circles.

**References**

Boyer, C. B. *A History of Mathematics.* New York: Wiley, p. 159, 1968.

Courant, R. and Robbins, H. ``Apollonius' Problem.'' §3.3 in
*What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.*
Oxford, England: Oxford University Press, pp. 117 and 125-127, 1996.

Dörrie, H. ``The Tangency Problem of Apollonius.'' §32 in
*100 Great Problems of Elementary Mathematics: Their History and Solutions.* New York: Dover, pp. 154-160, 1965.

Gauss, C. F. *Werke, Vol. 4.* New York: George Olms, p. 399, 1981.

Johnson, R. A. *Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle.*
Boston, MA: Houghton Mifflin, pp. 118-121, 1929.

Ogilvy, C. S. *Excursions in Geometry.* New York: Dover, pp. 48-51, 1990.

Pappas, T. *The Joy of Mathematics.* San Carlos, CA: Wide World Publ./Tetra, p. 151, 1989.

Simon, M. *Über die Entwicklung der Elementargeometrie im XIX Jahrhundert.* Berlin, pp. 97-105, 1906.

Weisstein, E. W. ``Plane Geometry.'' Mathematica notebook PlaneGeometry.m.

© 1996-9

1999-05-25