Ex-Center-Tetrahedra

Hidefumi Katsuura, San Jose State University

Abstract

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.