Ex-Center-Tetrahedra

Publication Date

1-1-2025

Document Type

Article

Publication Title

Journal for Geometry and Graphics

Volume

29

Issue

1

First Page

7

Last Page

17

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.

Keywords

deep interior of a tetrahedron, ex-center, ex-center-tetrahedron, ex-radius, ex-sphere, in-center, in-radius, in-sphere, isosceles MSC 2020: 51M04, tangled tetrahedron

Department

Mathematics and Statistics

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