Journal for Geometry and Graphics 29 (2025), No. 1, 007--017
Copyright by the authors licensed under CC BY SA 4.0
Ex-Center-Tetrahedra
Hidefumi Katsuura
San Jose State University, San Jose, U.S.A.
hidefumi.katsuura@sjsu.edu
The tetrahedron having ex-centers of a tetrahedron T as vertices is said to be the ex-center-tetrahedron of T, and let us denote it by T*. Theorem 1 shows that the ex-center-tetrahedron of a tetrahedron T and the tetrahedron that tangles T with respect to the in-center of T are the same. So T* is also used to denote the tetrahedron that tangles T with respect to the in-center of T. We define that a tetrahedron is weakly reversible if the sum of a pair of two face areas is equal to the sum of the other pair of two face areas. Theorem 2 shows that T is weakly reversible if and only if T and T* have the same volume. While T* may not be weakly reversible when T is weakly reversible, Theorem 3 shows that T* is reversible when T is reversible.
Keywords: Tangled tetrahedron, in-sphere, in-center, in-radius, ex-sphere, ex-center, ex-radius, ex-center-tetrahedron, deep interior of a tetrahedron, isosceles.
MSC: 51M04.
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