Non-Euclidean Geometries

Volume 581 of the series Mathematics and Its Applications pp 109-114

An Absolute Property of Four Mutually Tangent Circles

  • H.S.M. CoxeterAffiliated withUniversity of Toronto

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When Bolyai János was forty years old, Philip Beecroft discovered that any tetrad of mutually tangent circles determines a complementary tetrad such that each circle of either tetrad intersects three circles of the other tetrad orthogonally. By careful examination of a new proof of this theorem, one can see that it is absolute in Bolyai’s sense. Beecroft’s double-four of circles is seen to resemble Schläfli’s double-six of lines.