Three Collinear Points Generated by a Tetrahedron

Publication Date

1-1-2023

Document Type

Article

Publication Title

Journal for Geometry and Graphics

Volume

27

Issue

1

First Page

29

Last Page

37

Abstract

Let ABCD be a tetrahedron. For each point P inside of the tetrahedron ABCD, there is a unique set of points {E, F, G, H, I, J} such that (1) E, F, G, H, I, and J are points on the edges DA, AB, BC, CD, AC, and BD, respectively, and (2) the segments EG, FH, and IJ concur at P. If the three planes FGJ, GHI, EHJ, intersect, say at A∗, then we will prove that the three points A, P, A∗ are collinear. Let A′ be the intersection of the line AP and the plane BCD. If the points B∗, C∗, D∗ are defined similar to A∗, and if the points B′, C′, D′ are defined similar to A′, we will find the volume of the tetrahedra A∗B∗C∗D∗ and A′B′C′D′. We use barycentric coordinates to prove these results.

Keywords

barycentric coordinates, collinear points, tetrahedron

Department

Mathematics and Statistics

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