Isosceles Tetrahedron

A nonregular Tetrahedron in which each pair of opposite Edges are equal such that alltriangular faces are congruent. A Tetrahedron is isosceles Iff the sum of the face angles at eachVertex is 180°, and Iff its Insphere and Circumsphere areconcentric.

The only way for all the faces of a Tetrahedron to have the same Perimeter or to have the same Area is forthem to be fully congruent, in which case the tetrahedron is isosceles.

References

Brown, B. H. ``Theorem of Bang. Isosceles Tetrahedra.'' Amer. Math. Monthly 33, 224-226, 1926.

Honsberger, R. ``A Theorem of Bang and the Isosceles Tetrahedron.'' Ch. 9 in Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 90-97, 1976.

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