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A nonregular Tetrahedron in which each pair of opposite Edges are equal such that all triangular faces are congruent. A Tetrahedron is isosceles Iff the sum of the face angles at each Vertex is 180°, and Iff its Insphere and Circumsphere are concentric.
The only way for all the faces of a Tetrahedron to have the same Perimeter or to have the same Area is for them to be fully congruent, in which case the tetrahedron is isosceles.
See also Circumsphere, Insphere, Isosceles Triangle, Tetrahedron
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.