On Lexell's Theorem

@article{Maehara2017OnLT,
  title={On Lexell's Theorem},
  author={Hiroshi Maehara and Horst Martini},
  journal={The American Mathematical Monthly},
  year={2017},
  volume={124},
  pages={337 - 344}
}
Abstract Lexell's theorem states that two spherical triangles ABC and ABD have the same area if C and D lie on the same circular arc with endpoints A* and B*, which are the antipodal points of A and B, respectively. We present an elementary treatment for the case that the circular arc is a semicircle. In addition, a new proof of Lexell's theorem is presented, without using Girard's theorem. Finally, we give an improved version of Lexell's theorem in terms of the chord-tangent angle. 
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References

SHOWING 1-10 OF 19 REFERENCES
Spherical Triangles of Area π and Isosceles Tetrahedra
The final result will be the same. This shows that the ray emerging after three reflections is parallel to the original ray. Finally, we consider the special occurrence of a ray entering the cubeExpand
On the works of Euler and his followers on spherical geometry
We review and comment on some works of Euler and his followers on spherical geometry. We start by presenting some memoirs of Euler on spherical trigonometry. We comment on Euler's use of the methodsExpand
Modern Pure Solid Geometry
  • Nature
  • 1936
AbstractTHE scope of this book is more limited than its title indicates. The nine chapters deal respectively with preliminary ideas, trihedral angles, skew quadrilaterals, tetrahedra, transversals,Expand
On hyperbolic analogues of some classical theorems in spherical geometry
We provide hyperbolic analogues of some classical theorems in spherical geometry due to Menelaus, Euler, Lexell, Ceva and Lambert. Some of the spherical results are also made more precise.
INTRODUCTION TO GEOMETRY
This paper is an introduction to Riemannian and semi-Riemannian manifolds of constant sectional curvature. We will introduce the concepts of moving frames, curvature, geodesics and homogeneity on sixExpand
What Is Enlightenment
This is not a comprehensive article about what enlightenment is. I actually question what the enlightened state might be, and might not be, or in what context one might view it. For too long we haveExpand
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Mathématicien, physicien et Théoricien de la Musique. CNRSÉditions, Sciences de la Musique
  • SérieÉtudes
  • 2015
Lexell’s theorem, Normat
  • 2012
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