Casey's theorem in hyperbolic geometry

Release time：2024-04-11Views：10

Title:Casey's theorem in hyperbolic geometry

Speaker:Nikolay Abrosimov（Sobolev Institute of Mathematics, Novosibirsk）

Time:Monday, April 15, 2024, 14:30-15:30

Location：Mingde Building B201-1

Abstract：In 1881 Irish mathematician John Casey generalized Ptolemy’s theorem in the following way (see[1], p. 103).Casey’s theorem.Let circles O1, O2, O3, O4on a plane touch given circle O in vertices p1, p2, p3,p4of a convex quadrilateral. Denote by tijthe length of a common tangent of the circles Oiand Oj.If O separates Oiand Ojthen the internal tangent should be taken as tijelse the external tangentshould be taken. In both cases we assume that the tangents are exist. Then

In our paper [2], we produce hyperbolic version of Casey’s theorem.

Theorem 1.Let circles O1, O2, O3, O4on the hyperbolic plane H2touch given circle O in verticesp1, p2, p3, p4of a convex quadrilateral. Denote by tijthe length of a common tangent of the circlesOiand Oj. If O separates Oiand Ojthen the internal tangent should be taken as tijelse the externaltangent should be taken. In both cases we assume that the tangents are exist. Then

References:

[1] J. Casey, A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples, 5th. ed., Hodges, Figgis and Co., Dublin 1888.

[2] N.V. Abrosimov, L.A. Mikaiylova, Casey’s theorem in hyperbolic geometry // Siberian Electronic Mathematical Reports, 12 (2015), 354-360. DOI: https://doi.org/10.17377/semi.2015.12.029