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Journal for Geometry and Graphics 26 (2022), No. 2, 289--300
Copyright Heldermann Verlag 2022

Concurrent Segments in a Tetrahedron Ė Applications of Cevaís and Carnotís Theorems

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

Cevaís 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 Carnotís 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 Steiner-Routh theorem implies Carnotís theorem, and obtain the volumes of the two parts of a tetrahedron separated by a planar quadrilateral. Thirdly, we consider a special case of Carnotís theorem (or an extension of Varignonís 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.

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