![]() "A Unified Proof of Ceva and Menelaus' Theorems Using Projective Geometry" (PDF). The theorem also has a well-known generalization to spherical and hyperbolic geometry, replacing the lengths in the ratios with their sines and hyperbolic sines, respectively. The theorem has also been generalized to triangles on other surfaces of constant curvature. The analogue of the theorem for general polygons in the plane has been known since the early nineteenth century. Ceva's theorem can be obtained from it by setting the area equal to zero and solving. Routh's theorem gives the area of the triangle formed by three cevians in the case that they are not concurrent. Given a cycle of pairs of lobes, the product of the ratios of the volumes of the lobes in each pair is 1. Each of these points divides the face on which it lies into lobes. This point is the foot of a cevian that goes from the vertex opposite the k-face, in a ( k + 1)-face that contains it, through the point already defined on this ( k + 1)-face. Starting from a point in a simplex, a point is defined inductively on each k-face. Īnother generalization to higher-dimensional simplexes extends the conclusion of Ceva's theorem that the product of certain ratios is 1. Moreover, the intersection point of the cevians is the center of mass of the simplex. Then the cevians are concurrent if and only if a mass distribution can be assigned to the vertices such that each cevian intersects the opposite facet at its center of mass. ![]() Define a cevian of an n-simplex as a ray from each vertex to a point on the opposite ( n – 1)-face ( facet). The theorem can be generalized to higher-dimensional simplexes using barycentric coordinates. Ĭeva's theorem results immediately by taking the product of the three last equations. (The segments AD, BE, CF are known as cevians.) Then, using signed lengths of segments,Ī F ¯ F B ¯ ⋅ B D ¯ D C ¯ ⋅ C E ¯ E A ¯ = 1. Given a triangle △ ABC, let the lines AO, BO, CO be drawn from the vertices to a common point O (not on one of the sides of △ ABC), to meet opposite sides at D, E, F respectively. In Euclidean geometry, Ceva's theorem is a theorem about triangles. Ceva's theorem, case 1: the three lines are concurrent at a point O inside △ ABC Ceva's theorem, case 2: the three lines are concurrent at a point O outside △ ABC ![]() For other uses, see Ceva (disambiguation). ![]()
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