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Journal for Geometry and Graphics 27 (2023), No. 1, 029--037
Copyright Heldermann Verlag 2023

Three Collinear Points Generated by a Tetrahedron

Hidefumi Katsuura
San Jose State University, San Jose, U.S.A.

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.

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