Gyrotriangle Gyrocenters

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.


Scalar Equation Euclidean Geometry Tangent Point Hyperbolic Geometry Line Parameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 3.
    Bottema, O.: On the medians of a triangle in hyperbolic geometry. Can. J. Math. 10, 502–506 (1958) MathSciNetMATHCrossRefGoogle Scholar
  2. 11.
    Demirel, O., Soyturk, E.: The hyperbolic Carnot theorem in the Poincaré disc model of hyperbolic geometry. Novi Sad J. Math. 38(2), 33–39 (2008) MathSciNetMATHGoogle Scholar
  3. 26.
    Honsberger, R.: Episodes in Nineteenth and Twentieth Century Euclidean Geometry. New Mathematical Library, vol. 37, p. 174. Math. Assoc. Am., Washington (1995) MATHGoogle Scholar
  4. 28.
    Kelly, P.J., Matthews, G.: The Non-Euclidean, Hyperbolic Plane. Universitext, p. 333. Springer, New York (1981). Its structure and consistency MATHCrossRefGoogle Scholar
  5. 29.
    Kimberling, C.: Clark Kimberling’s Encyclopedia of Triangle Centers—ETC. (2010)
  6. 35.
    Maor, E.: Trigonometric Delights, p. 236. Princeton University Press, Princeton (1998) MATHGoogle Scholar
  7. 52.
    Sonmez, N.: A trigonometric proof of the Euler theorem in hyperbolic geometry. Algebras Groups Geom. 26(1), 75–79 (2009) MathSciNetGoogle Scholar
  8. 68.
    Vermeer, J.: A geometric interpretation of Ungar’s addition and of gyration in the hyperbolic plane. Topol. Appl. 152(3), 226–242 (2005) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Dept. Mathematics 2750North Dakota State UniversityFargoUSA

Personalised recommendations