### Abstract

A Steiner chain of length k consists of k circles tangent to two given non-intersecting circles (the parent circles) and tangent to each other in a cyclic pattern. The Steiner porism states that once a chain of k circles exists, there exists a 1-parameter family of such chains with the same parent circles that can be constructed starting with any initial circle tangent to the parent circles. What do the circles in these 1-parameter family of Steiner chains of length k have in common? We prove that the first k–1 moments of their curvatures remain constant within a 1-parameter family. For k = 3, this follows from the Descartes circle theorem. We extend our result to Steiner chains in spherical and hyperbolic geometries and present a related more general theorem involving spherical designs.

Original language | English (US) |
---|---|

Pages (from-to) | 238-248 |

Number of pages | 11 |

Journal | American Mathematical Monthly |

Volume | 127 |

Issue number | 3 |

DOIs | |

State | Published - Mar 15 2020 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Fingerprint Dive into the research topics of 'Descartes Circle Theorem, Steiner Porism, and Spherical Designs'. Together they form a unique fingerprint.

## Cite this

*American Mathematical Monthly*,

*127*(3), 238-248. https://doi.org/10.1080/00029890.2020.1690909