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Journal for Geometry and Graphics 08 (2004), No. 2, 163--169
Copyright Heldermann Verlag 2004

A Note on Bang's Theorem on Equifacial Tetrahedra

Mowaffaq Hajja
Dept. of Mathematics, Yarmouk University, Irbid, Jordan

Fathi Saidi
Dept. of Basic Sciences, University of Sharjah, P. O. Box 27272, Sharjah, United Arab Emirates

We give an analytic proof based on Pythagoras' theorem of a theorem of Bang stating that if the faces of a tetrahedron have equal areas then they are congruent. We also place Bang's theorem in the more general context that deals with the existence and uniqueness of a tetrahedron PABC having a given base ABC and having lateral faces of given areas. Our approach shows also how to construct counter-examples to Bang's statement in higher dimensions.

Keywords: Isosceles tetrahedron, equifacial tetrahedron, Bang's Theorem, regular simplex, barycentric coordinates, trilinear coordinates.

MSC: 51M20; 52B11

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