Journal for Geometry and Graphics 25 (2021), No. 2, 243--251
Copyright Heldermann Verlag 2021
Angles of Isosceles Tetrahedra
Hidefumi Katsuura
San Jose State University, San Jose, U.S.A.
hidefumi.katsuura@sjsu.edu
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.
Keywords: Isosceles tetrahedron, equifacial tetrahedron, dihedral angle, solid angle.
MSC: 51M04.
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