Geodesic
Three Collinear Points Generated by a Tetrahedron
Journal for geometry and graphics, Tome 27 (2023) no. 1, pp. 29-37.

Voir la notice de l'article provenant de la source Heldermann Verlag

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.
Classification : 51M04, 51M25
Mots-clés : Tetrahedron, collinear points, barycentric coordinates 
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     author = {H. Katsuura },
     title = {Three {Collinear} {Points} {Generated} by a {Tetrahedron}},
     journal = {Journal for geometry and graphics},
     pages = {29--37},
     publisher = {mathdoc},
     volume = {27},
     number = {1},
     year = {2023},
     url = {http://geodesic.mathdoc.fr/item/JGG_2023_27_1_JGG_2023_27_1_a2/}
}
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H. Katsuura . Three Collinear Points Generated by a Tetrahedron. Journal for geometry and graphics, Tome 27 (2023) no. 1, pp. 29-37. http://geodesic.mathdoc.fr/item/JGG_2023_27_1_JGG_2023_27_1_a2/