Generalizations of Casey’s Theorem for Higher Dimensions
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We give generalizations of Casey’s theorem and its converse for higher dimensions. We also present a multidimensional generalization for the problem of Apollonius. To do this we introduce a notion of ψ-tangent for a generalized k-sphere that touches a number of generalized n-balls in proper manner.
Keywords and phrasesCasey’s theorem Ptolemy’s theorem problem of Apollonius
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- 1.Ya. P. Ponarin, Elementary Geometry, Vol. 1: Planimetry (MCCME, Moscow, 2004) [in Russian].Google Scholar
- 4.W. J. McClelland and T. Preston, A Treatise on Spherical Trigonometry with Application to Spherical Geometry and Numerous Examples. Part II (Macmillian, London, 1886).Google Scholar
- 5.T. Kubota, “On the extended Ptolemy’s theorem in hyperbolic geometry,” Sci. Rep. Tohoku Univ., Ser. 1: Phys., Chem., Astron. 2, 131–156 (1912).Google Scholar
- 6.P. A. Shirokov, “Etudes on the Lobachevskii geometry,” Izv. Fiz.-Mat. Ob-va KGU, Ser. 2 24 (1), 26–32 (1924).Google Scholar
- 10.A. F. Berdon, The Geometry of Discrete Groups (Springer, New York, 1995).Google Scholar