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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.

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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