
Journal for Geometry and Graphics 27 (2023), No. 1, 029037 Copyright Heldermann Verlag 2023 Three Collinear Points Generated by a Tetrahedron Hidefumi Katsuura San Jose State University, San Jose, U.S.A. hidefumi.katsuura@sjsu.edu 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: Tetrahedron, collinear points, barycentric coordinates. MSC: 51M04; 51M25. [ Fulltextpdf (381 KB)] 