×

Lexell’s theorem via stereographic projection. (English) Zbl 1420.51014

Summary: Lexell’s theorem states that two spherical triangles \(\Delta{ABC}\) and \(\Delta{ABX}\) have the same area if \(C\) and \(X\) lie on the same circular arc with endpoints which are the antipodes of \(A\) and \(B\). In this note a proof is provided that exploits properties of the stereographic projection. We also prove Steiner’s theorem on areal bisectors of spherical triangles.

MSC:

51M09 Elementary problems in hyperbolic and elliptic geometries
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brannan, D.A., Esplen, M.F., Gray, J.J.: Geometry, 2nd edn. Cambridge University Press, New York (2012) · Zbl 1235.51001
[2] Casey, J.: A Treatise on Spherical Trigonometry, and Its Application to Geodesy and Astronomy with Numerous Examples. Hodges Figgis & Co., Dublin (1889)
[3] Donnay, J.D.H.: Spherical Trigonometry After the Cesàro Method. Interscience Publishers, New York (1945)
[4] Maehara, H., Martini, H.: On Lexell’s theorem. Am. Math. Mon. 124(4), 337-344 (2017) · Zbl 1391.51017
[5] Steiner, J.: Verwandlung und Teilung sphärischer Figuren durch construction. J. Reine Angew. Math. 2, 45-63 (1827) · ERAM 002.0045cj
[6] Van Brummelen, G.: Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry. Princeton University Press, Princeton (2013) · Zbl 1273.51003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.