thumb|upright=1.1|Ceva's theorem, case 1: the three lines are concurrent at a point inside

thumb|upright=1.1|Ceva's theorem, case 2: the three lines are concurrent at a point outside

In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle , let the lines be drawn from the vertices to a common point (not on one of the sides of ), to meet opposite sides at respectively. (The segments are known as cevians.) Then, using signed lengths of segments,

:<math>\frac{\overline{AF{\overline{FB \cdot \frac{\overline{BD{\overline{DC \cdot \frac{\overline{CE{\overline{EA = 1.</math>

In other words, the length is taken to be positive or negative according to whether is to the left or right of in some fixed orientation of the line. For example, is defined as having positive value when is between and 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 are chosen on respectively so that

: <math>\frac{\overline{AF{\overline{FB \cdot \frac{\overline{BD{\overline{DC \cdot \frac{\overline{CE{\overline{EA = 1,</math>

then 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 are the cevians of ), cevian triangle (the triangle is the cevian triangle of ); 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. From the transversal of triangle ,

: <math>\frac{\overline{AB{\overline{BF \cdot \frac{\overline{FO{\overline{OC \cdot \frac{\overline{CE{\overline{EA = -1</math>

and from the transversal of triangle ,

: <math>\frac{\overline{BA{\overline{AF \cdot \frac{\overline{FO{\overline{OC \cdot \frac{\overline{CD{\overline{DB = -1.</math>

The theorem follows by dividing these two equations.

The converse follows as a corollary.

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 -face. This point is the foot of a cevian that goes from the vertex opposite the -face, in a ()-face that contains it, through the point already defined on this ()-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.

See also

  • Projective geometry
  • Median (geometry) – an application
  • Circumcevian triangle
  • Menelaus's theorem
  • Triangle
  • Stewart's theorem
  • Cevian

References

Further reading

  • 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.
  • 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.