Central and Twin Tetrahedra

Publication Date

1-1-2023

Document Type

Article

Publication Title

Journal for Geometry and Graphics

Volume

27

Issue

2

First Page

119

Last Page

126

Abstract

Given a tetrahedron T, the tetrahedron T′ constructed by connecting the four centroids of its faces is called the central tetrahedron of T. A tetrahedron T can be inscribed in a parallelepiped W so that the edges of T are the diagonals of the faces of W. By drawing the remaining six diagonals on the faces of the parallelepiped W, we obtain a new tetrahedron T*, and call it the twin tetrahedron of T. Let S* and S*′ be the circumcenters of T* and T*′, respectively. We will prove that all tetrahedra T, T′, T*, and T*′ have the centroid in common, say P, and the five points S, S*′, P, S′, and S* are collinear in this order such that S# ′S»* = 2PS #»′, SP #» = 3PS #»′, SS #»′ = 2S# ′S»*, and SS #»* = 3S# ′S»*. Moreover, we prove that (1) T′ and T*′ are twins, and (2) if the tetrahedron T is orthocentric, then T, T′, T*, T*′ are orthocentric with orthocenters S*, S*′, S, and S′, respectively.

Keywords

central tetrahedron, centroid, circumcenter, orthocenter, orthocentric tetrahedron, twin tetrahedron

Department

Mathematics and Statistics

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