The theorems of Urquhart and Steiner-Lehmus in the Poincaré ball model of hyperbolic geometry

Authors

  • Oğuzhan Demirel Department of Mathematics, Faculty of Science and Arts, ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey Author
  • Emine Soytürk Seyrantepe Department of Mathematics, Faculty of Science and Arts, ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey Author

Keywords:

Möbius transformation, Gyrogroups, Hyperbolic geometry, Gyrovector spaces and hyperbolic trigonometry

Subjects:

51B10, 51M10, 30F45, 20N05

Abstract

In [Comput.~Math.~Appl. 41 (2001), 135–147],A.A. Ungar employs the Möbius gyrovector spaces for theintroduction of the hyperbolic trigonometry. This A.A. Ungar'swork, plays a major role in translating some theorems in Euclideangeometry to corresponding theorems in hyperbolic geometry. In thispaper we present (i)~the hyperbolic Breusch's lemma, (ii)~ thehyperbolic Urquhart's theorem, and (iii)~ the hyperbolicSteiner-Lehmus theorem in the Poincaré ball model ofhyperbolic geometry by employing results from A.A. Ungar's work.

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Published

2011-10-15

Issue

Vol. 63 No. 4 (2011): Vol. 63, Issue 4

Section

Articles

License

Copyright (c) 2011 Authors retain copyright to their work.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.