Ceva's theorem is a theorem about triangles in plane geometry. Given a triangle ABC, let the lines AO, BO and 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 and F respectively. (The segments AD, BE, and CF are known as cevians.) Then, using signed lengths of segments,
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 and F are chosen on BC, AC and AB respectively so that
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.
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 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.
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
(Replace the minus with a plus if A and O are on opposite sides of BC.) Similarly,
Multiplying these three equations gives
The theorem can also be proven easily using Menelaus' theorem. From the transversal BOE of triangle ACF,
and from the transversal AOD of triangle BCF,
The theorem follows by dividing these two equations.
The converse follows as a corollary. Let D, E and F be given on the lines BC, AC and AB so that the equation holds. Let AD and BE meet at O and let F′ be the point where CO crosses AB. Then by the theorem, the equation also holds for D, E and F′. Comparing the two,
But at most one point can cut a segment in a given ratio so F=F′.
Using barycentric coordinates
for every point X (for the definition of this arrow notation and further details, see Affine space).
For Ceva's theorem, the point O is supposed to not belong to any line passing through two vertices of the triangle. This implies that
If one takes for X the intersection F of the lines AB and OC (see figures), the last equation may be rearranged into
The left-hand side of this equation is a vector that has the same direction as the line CF, and the right-hand side has the same direction as the line AB. These lines have different directions since A, B, and C are not collinear. 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.
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 simplexs 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.
The analogue of the theorem for general polygons in the plane has been known since the early nineteenth century. The theorem has also been generalized to triangles on other surfaces of constant curvature.
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.
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- Menelaus and Ceva at MathPages
- Derivations and applications of Ceva's Theorem at cut-the-knot
- Trigonometric Form of Ceva's Theorem at cut-the-knot
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