Ceva's theorem

In other words, the length XY is taken to be positive or negative according to whether X is to the left or right of Y in some fixed orientation of the line. For example, AF / FB is defined as having positive value when F is between A and B and negative otherwise.


Ceva's theorem is a theorem of affine geometry, in the sense that it may be stated and proved without using the concepts of angles, areas, and lengths (except for the ratio of the lengths of two line segments that are collinear). It is therefore true for triangles in any affine plane over any field.


A slightly adapted converse is also true: If points D, E, F are chosen on BC, AC, AB respectively so that


then AD, BE, CF are concurrent, or all three parallel. The converse is often included as part of the theorem.


The theorem is often attributed to Giovanni Ceva, who published it in his 1678 work De lineis rectis. But it was proven much earlier by Yusuf Al-Mu'taman ibn Hűd, an eleventh-century king of Zaragoza.^[1]


Associated with the figures are several terms derived from Ceva's name: cevian (the lines AD, BE, CF are the cevians of O), cevian triangle (the triangle △DEF is the cevian triangle of O); cevian nest, anticevian triangle, Ceva conjugate. (Ceva is pronounced Chay'va; cevian is pronounced chev'ian.)


The theorem is very similar to Menelaus' theorem in that their equations differ only in sign. By re-writing each in terms of cross-ratios, the two theorems may be seen as projective duals.^[2]

Generalizations[edit]

The theorem can be generalized to higher-dimensional simplexes using barycentric coordinates. Define a cevian of an n-simplex as a ray from each vertex to a point on the opposite (n – 1)-face (facet). 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. Moreover, the intersection point of the cevians is the center of mass of the simplex.^[6][7]


Another generalization to higher-dimensional simplexes extends the conclusion of Ceva's theorem that the product of certain ratios is 1. Starting from a point in a simplex, a point is defined inductively on each k-face. 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. Each of these points divides the face on which it lies into lobes. Given a cycle of pairs of lobes, the product of the ratios of the volumes of the lobes in each pair is 1.^[8]


Routh's theorem gives the area of the triangle formed by three cevians in the case that they are not concurrent. Ceva's theorem can be obtained from it by setting the area equal to zero and solving.


The analogue of the theorem for general polygons in the plane has been known since the early nineteenth century.^[9] The theorem has also been generalized to triangles on other surfaces of constant curvature.^[10]


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.