# The Circles of Descartes

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In a circle, the curvature (or bend) is the reciprocal of the radius, . (When measured from the inside, the curvature is negative.) Four mutually tangent circles have the property . This is the Descartes circle theorem. With the centers in the complex plane (), the following formula holds:

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Contributed by: Ed Pegg Jr (December 2008)

Open content licensed under CC BY-NC-SA

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References

[1] J. C. Lagarias, C. L. Mallows, A. R. Wilks, "Beyond the Descartes Circle Theorem," http://arxiv.org/abs/math.MG/0101066, Jan. 9, 2001.

[2] D. Austin, "When Kissing Involves Trigonometry," http://www.ams.org/featurecolumn/archive/kissing.html, March 2006.

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