
Journal for Geometry and Graphics 26 (2022), No. 2, 289300 Copyright Heldermann Verlag 2022 Concurrent Segments in a Tetrahedron – Applications of Ceva’s and Carnot’s Theorems Hidefumi Katsuura San Jose State University, San Jose, U.S.A. hidefumi.katsuura@sjsu.edu Ceva’s theorem is about concurrence of three segments on a triangle with an affine ratio. Among the several theorems named after him, we are interested in Carnot’s theorem that relates the concurrence of two segments in a skew quadrilateral in space, again, with an affine ratio. First, we apply these theorems to obtain a theorem on the concurrence of seven segments in a tetrahedron. Secondly, we show that the SteinerRouth theorem implies Carnot’s theorem, and obtain the volumes of the two parts of a tetrahedron separated by a planar quadrilateral. Thirdly, we consider a special case of Carnot’s theorem (or an extension of Varignon’s theorem) to determine when four points on a skew quadrilateral are to form a parallelogram. Finally, we give a new characterization of the centroid of a tetrahedron. Keywords: Tetrahedron, Ceva's theorem, Carnot's theorem, concurrence, affine ratio, centroid. MSC: 51M04; 51M25. [ Fulltextpdf (2050 KB)] 