Part of the
Fundamental Theories of Physics
book series (FTPH, volume 166)
Interest in triangle centers has long history, the classical ones being the triangle centroid, orthocenter, incenter and circumcenter. A list of more than 3000 triangle centers is found in Kimberling (Clark Kimberling’s Encyclopedia of Triangle Centers—ETC, 2010). Hyperbolic triangles and their centers are of interest as well (Bottema in Can. J. Math. 10:502–506, 1958; Vermeer in Topol. Appl. 152(3):226–242, 2005; Demirel and Soyturk in Novi Sad J. Math. 38(2):33–39, 2008; Sonmez in Algebras Groups Geom. 26(1):75–79, 2009). The special relativistic approach of this book enables hyperbolic triangle centers to be determined along with relationships between them.
The hyperbolic triangle circumcenter, incenter and orthocenter are called, in gyrolanguage, the gyrotriangle circumgyrocenter, ingyrocenter and orthogyrocenter, respectively. These gyrocenters are determined in this chapter in terms of their gyrobarycentric coordinate representations with respect to the vertices of their reference gyrotriangles.
KeywordsScalar Equation Euclidean Geometry Tangent Point Hyperbolic Geometry Line Parameter
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Kimberling, C.: Clark Kimberling’s Encyclopedia of Triangle Centers—ETC. http://faculty.evansville.edu/ck6/encyclopedia/ETC.html
Maor, E.: Trigonometric Delights, p. 236. Princeton University Press, Princeton (1998)
Sonmez, N.: A trigonometric proof of the Euler theorem in hyperbolic geometry. Algebras Groups Geom. 26
(1), 75–79 (2009)
Vermeer, J.: A geometric interpretation of Ungar’s addition and of gyration in the hyperbolic plane. Topol. Appl. 152
(3), 226–242 (2005)
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