塞瓦定理

它的逆定理同样成立：若 D、 E、 F 分别在 （公式略） 的边 BC、 CA、 AB 或其延长线上（都在边上或有两点在延长线上），且满足 BD/DC· CE/EA· AF/FB=1，

则直线 AD、 BE、 CF 共点或彼此平行（于无限远处共点）。当 AD、 BE、 CF 中的任意两直线交于一点时，则三直线共点；当 AD、 BE、 CF 中的任意两直线平行时，则三直线平行。

它最先由意大利数学家乔瓦尼·塞瓦证明，因而得名。此定理又译西瓦定理或帅氏定理。

证明

• 设 h=A 到 BC 的距离 （公式略） 的面积 = 1/2· h· BD
• △ （公式略） 的面积 = 1/2· h· DC
• ∴ （公式略） 的面积 / （公式略） 的面积 = BD/DC
• ∵ （公式略）
• 由等比性质, （公式略）
• 同理 （公式略）
• ∴ （公式略）

证毕。

推论：角平分线定理

在三角形 ABC 中， ∠ A 的角平分线交 BC 于 D， DB/DC=AB/AC。

In Euclidean geometry, Ceva's theorem is a theorem about triangles. 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. (The segments AD, BE, CF are known as cevians.) Then, using signed lengths of segments, AF/FB· BD/DC· CE/EA=1.

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 AF/FB· BD/DC· CE/EA=1,

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. Ibn Hűd's work, however, had fallen into oblivion, and was rediscovered only in 1985.

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.

Proofs

Several proofs of the theorem have been created.

Two proofs are given in the following.

The first one is very elementary, using only basic properties of triangle areas. However, several cases have to be considered, depending on the position of the point O.

The second proof uses barycentric coordinates and vectors, but is not case dependent. Moreover, it works in any affine plane over any field.

Using triangle areas

First, the sign of the left-hand side is positive since either all three of the ratios are positive, the case where O is inside the triangle (upper diagram), or one is positive and the other two are negative, the case O is outside the triangle (lower diagram shows one case).

To check the magnitude, note that the area of a triangle of a given height is proportional to its base. So （公式略）

• Therefore, （公式略）
• (Replace the minus with a plus if A and O are on opposite sides of BC.)
• Similarly, （公式略）
• and （公式略）
• Multiplying these three equations gives |AF/FB· BD/DC· CE/EA|=1,

as required.

The theorem can also be proven easily using Menelaus's theorem. From the transversal BOE of triangle △ACF, AB/BF· FO/OC· CE/EA=-1

and from the transversal AOD of triangle △BCF, BA/AF· FO/OC· CD/DB=-1.

The theorem follows by dividing these two equations.

Generalizations

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.

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.

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.