Geodesic
Angles of Isosceles Tetrahedra
Journal for geometry and graphics, Tome 25 (2021) no. 2, pp. 243-251.

Voir la notice de l'article provenant de la source Heldermann Verlag

We give three new characteristics of an isosceles tetrahedron. These three characteristics are: (i) the sum of cosines of dihedral angles of a tetrahedron at each vertex is 1, (ii) the opposing dihedral angles of a tetrahedron are pairwise same, and (iii) all four solid angles of a tetrahedron are the same. It is known that "isosceles" implies (ii) and (iii), but we think the converse of these and (i) are new. The statement (iii) suggests that a solid angle may determine an isosceles tetrahedron uniquely up to a similarity. However, we give an example to show that this is not the case unless it is a regular tetrahedron. And finally, we obtain a trigonometric identity from an isosceles tetrahedron. We use a theorem on a spherical triangle.
Classification : 51M04
Mots-clés : Isosceles tetrahedron, equifacial tetrahedron, dihedral angle, solid angle 
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     title = {Angles of {Isosceles} {Tetrahedra}},
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     number = {2},
     year = {2021},
     url = {http://geodesic.mathdoc.fr/item/JGG_2021_25_2_JGG_2021_25_2_a7/}
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H. Katsuura . Angles of Isosceles Tetrahedra. Journal for geometry and graphics, Tome 25 (2021) no. 2, pp. 243-251. http://geodesic.mathdoc.fr/item/JGG_2021_25_2_JGG_2021_25_2_a7/