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A Straightforward Proof of Descartes’s Circle Theorem

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Notes

  1. All the figures were produced by the author.

  2. This is a problem at which a countess tried her hand, Countess Skorzewska [1, p. 308]. Lambert describes her in his correspondence as a learned Polish lady and a great lover (“Liebhaber”) of the mathematical sciences.

  3. As Descartes writes in his letter to Elisabeth dated November 29, 1643 [5], “But this route seems to me to lead to so many superfluous multiplications that I would not want to undertake to solve them in three months.”

References

  1. Lambert, J. H. Deutscher gelehrter Briefwechsel, Band I. Berlin, 1781.

  2. Coxeter, H. S. M. The Problem of Apollonius. Amer. Math. Monthly 75:1 (1968), 5–15.

  3. Coxeter, H. S. M. Loxodromic Sequences of Tangent Spheres. Aequationes Math. 1 (1968), 104–121.

  4. Descartes, R. Oeuvres. Published by Ch. Adam and Paul Tannery; Correspondance, Vol. IV (July 1643 to April 1647). Paris, Léopold Cerf, 1901, pp. 37, 49.

  5. Shapiro, L. The Correspondence between Princess Elisabeth of Bohemia and René Descartes. University of Chicago Press, Chicago, 2007.

  6. Soddy, F. The Kiss Precise. Nature 137 (1936), 1021.

  7. Nature 139 (1937), 62.

  8. Lagarias, J. C., Mallows, C. L., and Wilks A. R. Beyond the Descartes Circle Theorem. Amer. Math. Monthly 109:4 (2002), 338–361.

  9. Steiner, J. Fortsetzung der geometrischen Betrachtungen, J. Reine Angew. Math. 1 (1826), 252–288.

  10. Pedoe, D. On a Theorem in Geometry. Amer. Math. Monthly 74:6 (1967), 627–640.

  11. Nelsen, R. B. Heron’s Formula via Proofs Without Words. College Math. J. 32:4 (2001), 290–292.

  12. Sarnak, P. Integral Apollonian Packings. Amer. Math. Monthly 118:4 (2011), 291–306.

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Acknowledgments

I want to thank Damaris Schindler, of Utrecht University, for bringing me back to this old favorite of mine.

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  1. Faculty of Applied Engineering, UAntwerpen, Groenenborgerlaan 171, 2020, Antwerp, Belgium

    Paul Levrie

  2. Department of Computer Science, KU Leuven, Celestijnenlaan 200A, P.O. Box 2402, 3001, Heverlee (Leuven), Belgium

    Paul Levrie

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Levrie, P. A Straightforward Proof of Descartes’s Circle Theorem. Math Intelligencer 41, 24–27 (2019). https://doi.org/10.1007/s00283-019-09883-x

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