Ceva's theorem

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).

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 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.

The second proof uses barycentric coordinates and vectors, but is somehow[vague] more natural and not case dependent.

To check the magnitude, note that the area of a triangle of a given height is proportional to its base.

For Ceva's theorem, the point O is supposed to not belong to any line passing through two vertices of the triangle.

It follows that the two members of the equation equal the zero vector, and It follows that where the left-hand-side fraction is the signed ratio of the lengths of the collinear line segments AF and FB.

The same reasoning shows Ceva's theorem results immediately by taking the product of the three last equations.

Define a cevian of an n-simplex as a ray from each vertex to a point on the opposite (n – 1)-face (facet).

[6][7] Another generalization to higher-dimensional simplexes extends the conclusion of Ceva's theorem that the product of certain ratios is 1.

[8] Routh's theorem gives the area of the triangle formed by three cevians in the case that they are not concurrent.

The analogue of the theorem for general polygons in the plane has been known since the early nineteenth century.