A Straightforward Proof of Descartes’s Circle Theorem

  • Paul LevrieEmail author

There are probably not many formulas in mathematics in the discovery of which a princess was instrumental and that are described in a poem written by a Nobel laureate (in Chemistry!). Yet such is the case for what is now known as Descartes’s circle formula.

Let \(C_1\)



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


  1. [1]
    Lambert, J. H. Deutscher gelehrter Briefwechsel, Band I. Berlin, 1781.Google Scholar
  2. [2]
    Coxeter, H. S. M. The Problem of Apollonius. Amer. Math. Monthly 75:1 (1968), 5–15.Google Scholar
  3. [3]
    Coxeter, H. S. M. Loxodromic Sequences of Tangent Spheres. Aequationes Math. 1 (1968), 104–121.Google Scholar
  4. [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.Google Scholar
  5. [5]
    Shapiro, L. The Correspondence between Princess Elisabeth of Bohemia and René Descartes. University of Chicago Press, Chicago, 2007.Google Scholar
  6. [6]
    Soddy, F. The Kiss Precise. Nature 137 (1936), 1021.Google Scholar
  7. [7]
    Nature 139 (1937), 62.Google Scholar
  8. [8]
    Lagarias, J. C., Mallows, C. L., and Wilks A. R. Beyond the Descartes Circle Theorem. Amer. Math. Monthly 109:4 (2002), 338–361.Google Scholar
  9. [9]
    Steiner, J. Fortsetzung der geometrischen Betrachtungen, J. Reine Angew. Math. 1 (1826), 252–288.Google Scholar
  10. [10]
    Pedoe, D. On a Theorem in Geometry. Amer. Math. Monthly 74:6 (1967), 627–640.Google Scholar
  11. [11]
    Nelsen, R. B. Heron’s Formula via Proofs Without Words. College Math. J. 32:4 (2001), 290–292.Google Scholar
  12. [12]
    Sarnak, P. Integral Apollonian Packings. Amer. Math. Monthly 118:4 (2011), 291–306.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Applied EngineeringUAntwerpenAntwerpBelgium
  2. 2.Department of Computer ScienceKU LeuvenHeverlee (Leuven)Belgium

Personalised recommendations