Concurrent Segments in a Tetrahedron – Applications of Ceva’s and Carnot’s Theorems
Journal for Geometry and Graphics
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.
Mathematics and Statistics
Hidefumi Katsuura. "Concurrent Segments in a Tetrahedron – Applications of Ceva’s and Carnot’s Theorems" Journal for Geometry and Graphics (2022): 289-300.