MATEMATIČKI VESNIK
МАТЕМАТИЧКИ ВЕСНИК



MATEMATIČKI VESNIK
The theorems of Urquhart and Steiner-Lehmus in the Poincaré ball model of hyperbolic geometry
O\u{g}uzhan Demirel and Emine Soytürk Seyrantepe

Abstract

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

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Keywords: Möbius transformation; Gyrogroups; Hyperbolic geometry; Gyrovector spaces and hyperbolic trigonometry.

MSC: 51B10, 51M10, 30F45, 20N05

Pages:  263$-$274     

Volume  63 ,  Issue  3 ,  2011