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13,123 entries Last updated: Tue Aug 7 2012

Created, developed, and nurtured by Eric Weisstein at Wolfram Research

Geometry > Plane Geometry > Triangles > Triangle Circles >

Interactive Entries > Interactive Demonstrations >

Soddy Circles

Given three noncollinear points, construct three tangent circles such that one is centered at each point and the circles are pairwise tangent to one another. Then there exist exactly two nonintersecting circles that are tangent to all three circles. These are called the inner and outer Soddy circles, and their centers are called the inner and outer Soddy centers , respectively.

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

(1)

where are the so-called bends, defined as the signed curvatures of the circles. If the contacts are all external, the signs are all taken as positive, whereas if one circle surrounds the other three, the sign of this circle is taken as negative (Coxeter 1969). Using the quadratic formula to solve for , expressing in terms of radii instead of curvatures, and simplifying gives

(2)

Here, the negative solution corresponds to the outer Soddy circle and the positive one to the inner Soddy circle.

The three lines through opposite points of tangency of any four mutually tangent circles are coincident, where "opposite" means here that the two circles determining one point of tangency are distinct from the two circles determining the other (Eppstein 2001). This fact gives rise to the first and second Eppstein points.

This formula is called the Descartes circle theorem since it was known to Descartes. Soddy extended the result to tangent spheres, and Gosper has further extended the result to mutually tangent -dimensional hyperspheres.

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

(3)

where is the curvature of the central circle,

(4)

and

(5)

For , this simplifies to the Descartes circle theorem

(6)

SEE ALSO:Apollonian Gasket, Apollonius Circle, Apollonius' Problem, Arbelos, Bend, Bowl of Integers, Circumcircle, Descartes Circle Theorem, Excentral Triangle, Four Coins Problem, Hart's Theorem, Inner Soddy Center, Inner Soddy Circle, Malfatti Circles, Outer Soddy Center, Outer Soddy Circle, Pappus Chain, Soddy Centers, Soddy Triangles, Sphere Packing, Steiner Chain, Tangent Circles, Tangent Spheres

REFERENCES:Berger, M.; Pansu, P.; Berry, J.-P.; and Saint-Raymond, X. Problems in Geometry. New York: Springer-Verlag, 1984.

Boyd, D. W. "The Sequence of Radii of the Apollonian Packing." Math. Comput. 39, 249-254, 1982.

Coxeter, H. S. M. "The Problem of Apollonius." Amer. Math. Monthly 75, 5-15, 1968.

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

Dergiades, N. "The Soddy Circles." Forum Geometricorum 7, 191-197, 2007. http://forumgeom.fau.edu/FG2007volume7/FG200726index.html.

Dodds, P. S. and Weitz, J. S. "Packing-Limited Growth." Phys. Rev. E 65, 056108, 2002.

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

Eppstein, D. "Tangent Spheres and Triangle Centers." Amer. Math. Monthly 108, 63-66, 2001.

Fillmore, J. P. and Paluszny, M. Seminarberichte Mathematik Fernuniversität Hagen 62, 45, 1997.

Gardner, M. Fractal Music, Hypercards, and More: Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, 1992.

Gosper, R. W. "Soddy's Theorem on Mutually Tangent Circles, Generalized to Dimensions." http://www.ippi.com/rwg/Sodddy.htm.

Apollonian GaskMichael Schreiber

The Circles of Descartes Ed Pegg Jr

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Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H. "Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group." 15 Jul 2004. http://arxiv.org/abs/math/0010298.

Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H. "Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings." 15 Jul 2004. http://arxiv.org/abs/math/0010302.

Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H. "Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions." 16 Jun 2004. http://arxiv.org/abs/math/0010324.

Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H. "Apollonian Circle Packings: Number Theory." J. Number Th. 100, 1-45, 2003.

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

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

"The Kiss Precise." Nature 139, 62, 1937.

Martini, H. In Geometrie und ihre Anwendungen in Kunst, Natur und Technik (Ed. O. Giering and J. Hoschek). Munich, Germany: Carl Hanser, 1994.

Oldknow, A. "Computer Aided Research into the Geometry of the Triangle." Math. Gaz. 79, 263-274, 1995.

Söderberg, B. Phys. Rev. A 46, 1859, 1992.

Soddy, F. "The Kiss Precise." Nature 137, 1021, 1936.

Study, E. Math. Ann. 49, 497, 1897.

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

Veldkamp, G. R. "A Theorem Concerning Soddy-Circles." Elem. Math. 21, 15-17, 1966.

Veldkamp, G. R. "The Isoperimetric Point and the Point(s) of Equal Detour." Amer. Math. Monthly 92, 546-558, 1985.

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 4-5, 1991.

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