Geodesic
Concurrent Segments in a Tetrahedron – Applications of Ceva’s and Carnot’s Theorems
Journal for geometry and graphics, Tome 26 (2022) no. 2, pp. 289-3
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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 Steiner-Routh 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.
Classification : 51M04, 51M25
Mots-clés : Tetrahedron, Ceva's theorem, Carnot's theorem, concurrence, affine ratio, centroid 
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     author = {H. Katsuura},
     title = {Concurrent {Segments} in a {Tetrahedron} \^a€“ {Applications} of {Ceva\^a€™s} and {Carnot\^a€™s} {Theorems}},
     journal = {Journal for geometry and graphics},
     pages = {289--3},
     year = {2022},
     volume = {26},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JGG_2022_26_2_JGG_2022_26_2_a6/}
}
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H. Katsuura. Concurrent Segments in a Tetrahedron – Applications of Ceva’s and Carnot’s Theorems. Journal for geometry and graphics, Tome 26 (2022) no. 2, pp. 289-3. http://geodesic.mathdoc.fr/item/JGG_2022_26_2_JGG_2022_26_2_a6/