# Ceva's theorem

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**Ceva's theorem** is a theorem about triangles in Euclidean plane geometry. Given a triangle *ABC*, let the lines *AO*, *BO* and *CO* be drawn from the vertices to a common point *O* 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 *AB* is taken to be positive or negative according to whether *A* is to the left or right of *B* 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.

The converse is also true: If points *D*, *E* and *F* are chosen on *BC*, *AC* and *AB* respectively so that

then *AD*, *BE* and *CF* are concurrent. 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.

## Proof of the theorem

(Here directed segments are not used, except in the case of proving using Menelaus' Theorem)

A standard proof is as follows;^{[2]} Posamentier and Salkind^{[3]}^{:177-180} give four proofs.

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

Therefore,

(Replace the minus with a plus if *A* and *O* are on opposite sides of *BC*.)
Similarly,

and

Multiplying these three equations gives

as required.

The theorem can also be proven easily using Menelaus' theorem.^{[4]} 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.^{[5]} 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*′.

## 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. (Landy. See Wernicke for an earlier result.)

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 Template:Harv. The theorem has also been generalized to triangles on other surfaces of constant curvature (Masal'tsev 1994).

## See also

- Projective geometry
- Median (geometry) - an application

## References

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## External links

- 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
- Glossary of Encyclopedia of Triangle Centers includes definitions of cevian triangle, cevian nest, anticevian triangle, Ceva conjugate, and cevapoint
- Conics Associated with a Cevian Nest, by Clark Kimberling
*Ceva's Theorem*by Jay Warendorff, Wolfram Demonstrations Project.- Weisstein, Eric W., "Ceva's Theorem",
*MathWorld*. - Experimentally finding the centroid of a triangle with different weights at the vertices: a practical application of Ceva's theorem at Dynamic Geometry Sketches, an interactive dynamic geometry sketch using the gravity simulator of Cinderella.