Geodesic
Casey's theorem in hyperbolic geometry
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 354-360.

Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain a hyperbolic version of Casey's theorem. A similar result is obtained in spherical geometry as well.
Keywords: Casey's theorem, Ptolemy's theorem, hyperbolic plane, spherical plane. 
@article{SEMR_2015_12_a40,
     author = {N. V. Abrosimov and L. A. Mikaiylova},
     title = {Casey's theorem in hyperbolic geometry},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {354--360},
     publisher = {mathdoc},
     volume = {12},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2015_12_a40/}
}
TY  - JOUR
AU  - N. V. Abrosimov
AU  - L. A. Mikaiylova
TI  - Casey's theorem in hyperbolic geometry
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2015
SP  - 354
EP  - 360
VL  - 12
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2015_12_a40/
LA  - en
ID  - SEMR_2015_12_a40
ER  - 
%0 Journal Article
%A N. V. Abrosimov
%A L. A. Mikaiylova
%T Casey's theorem in hyperbolic geometry
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2015
%P 354-360
%V 12
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2015_12_a40/
%G en
%F SEMR_2015_12_a40
N. V. Abrosimov; L. A. Mikaiylova. Casey's theorem in hyperbolic geometry. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 354-360. http://geodesic.mathdoc.fr/item/SEMR_2015_12_a40/

[1] D. V. Alekseevskii, E. B. Vinberg, A. S. Solodovnikov, Geometry, v. II, Encycl. Math. Sci., 29, Spaces of Constant Curvature, Springer, Berlin, 1993, 138 pp. | MR

[2] G. A. Baigonakova, A. D. Mednykh, “On Bretschneider's formula for a spherical quadrilateral”, Mat. Zamet. YaGU, 19:2 (2012), 3–11 | Zbl

[3] G. A. Baigonakova, A. D. Mednykh, “On Bretschneider's formula for a hyperbolic quadrilateral”, Mat. Zamet. YaGU, 19:2 (2012), 12–19 | Zbl

[4] H. S. M. Coxeter, S. L. Greitzer, Ptolemy's Theorem and its Extensions, Math. Assoc. Amer., Washington, D.C., 1967

[5] J. Casey, A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples, 5th. ed., Hodges, Figgis and Co., Dublin, 1888

[6] T. Kubota, “On the extended Ptolemy's theorem in hyperbolic geometry”, Science reports of the Tohoku University. Ser. 1: Physics, chemistry, astronomy, 2 (1912), 131–156

[7] M. P. Limonov, “On some aspects of a hyperbolic tangential quadrilateral”, Sib. Èlektron. Mat. Izv., 10 (2013), 454–463 | MR

[8] W. J. M'Clelland, T. Preston, A Treatise on Spherical Trigonometry with application to Spherical Geometry and Numerous Examples, v. II, Macmillian and So., London, 1886

[9] A. D. Mednykh, “Brahmagupta formula for cyclic quadrilaterals in the hyperbolic plane”, Sib. Èlektron. Mat. Izv., 9 (2012), 247–255 | MR

[10] D. Yu. Sokolova, “On trapezoid area on the Lobachevskii plane”, Sib. Èlektron. Mat. Izv., 9 (2012), 256–260 | MR

[11] J. E. Valentine, “An analogue of Ptolemy's theorem and its converse in hyperbolic geometry”, Pacific J. Math., 34:3 (1970), 817–825 | DOI | MR | Zbl

[12] L. Wimmer, “Cyclic polygons in non-Euclidean geometry”, Elem. Math., 66:2 (2011), 74–82 | DOI | MR | Zbl