Given three distinct points , , and , let three Circles be drawn, one centered about each point and each one tangent to the other two. Call the Radii (, , ). Then the Circles satisfy

(1) |

(2) |

(3) |

Solving for the Radii then gives

(4) | |||

(5) | |||

(6) |

The above Triangle has sides , , and , and Semiperimeter

(7) |

(8) |

(9) |

(10) |

(11) | |||

(12) | |||

(13) |

As can be seen from the first figure, there exist exactly two nonintersecting Circles which are Tangent to all three Circles. These are called the inner and outer Soddy circles ( and , respectively), and their centers are called the inner and outer Soddy Points.

The inner Soddy circle is the solution to the Four Coins Problem. The center of the inner Soddy circle is the Equal Detour Point, and the center of the outer Soddy circle is the Isoperimetric Point (Kimberling 1994).

Frederick Soddy (1936) gave the Formula for finding the Radii of the Soddy circles () given the
Radii (, 2, 3) of the other three. The relationship is

(14) |

(15) |

This Formula is called the Descartes Circle Theorem since it was known to Descartes. However, Soddy
also extended it to Spheres. Gosper has further extended the result to mutually tangent -D
Hyperspheres, whose Curvatures satisfy

(16) |

(17) |

For (at least) and 3, the Radical equals

(18) |

Bellew has derived a generalization applicable to a Circle surrounded by Circles which are, in turn, circumscribed by another Circle. The relationship is

(19) |

(20) |

**References**

Coxeter, H. S. M. *Introduction to Geometry, 2nd ed.* New York: Wiley, pp. 13-14, 1969.

Elkies, N. D. and Fukuta, J. ``Problem E3236 and Solution.'' *Amer. Math. Monthly* **97**, 529-531, 1990.

Kimberling, C. ``Central Points and Central Lines in the Plane of a Triangle.'' *Math. Mag.* **67**, p. 181, 1994.

``The Kiss Precise.'' *Nature* **139**, 62, 1937.

Soddy, F. ``The Kiss Precise.'' *Nature* **137**, 1021, 1936.

Vandeghen, A. ``Soddy's Circles and the De Longchamps Point of a Triangle.'' *Amer. Math. Monthly* **71**, 176-179, 1964.

© 1996-9

1999-05-26