In Euclidean geometry, Ceva's theorem is a theorem about

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

s. Given a triangle , let the

lines Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...

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, :

\frac  \cdot \frac \cdot \frac = 1.

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 segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

s that are collinear). It is therefore true for triangles in any

affine plane In geometry, an affine plane is a two-dimensional affine space. Examples Typical examples of affine planes are * Euclidean planes, which are affine planes over the reals equipped with a metric, the Euclidean distance. In other words, an affine pl ...

over any

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

. A slightly adapted converse is also true: If points are chosen on respectively so that :

\frac  \cdot \frac \cdot \frac = 1,

then are

concurrent Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to: Law * Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea'' * Concurring opinion (also called a "concurrence"), a ...

, or all three

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of IBM ...

. The converse is often included as part of the theorem. The theorem is often attributed to

Giovanni Ceva Giovanni Ceva (September 1, 1647 – May 13, 1734) was an Italian mathematician widely known for proving Ceva's theorem in elementary geometry. His brother, Tommaso Ceva was also a well-known poet and mathematician. Life Ceva received his educat ...

, 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 Zaragoza, also known in English as Saragossa,''Encyclopædia Britannica'"Zaragoza (conventional Saragossa)" is the capital city of the Zaragoza Province and of the autonomous community of Aragon, Spain. It lies by the Ebro river and its tributari ...

. 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 given. 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 . The second proof uses barycentric coordinates and vectors, but is somehow more natural and not case dependent. Moreover, it works in any

over any

Using triangle areas

First, the sign of the

left-hand side In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric.collinear, and a point , that belongs to the same

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * Planes (gen ...

, the barycentric coordinates of with respect of are the unique three numbers

\lambda_A, \lambda_B, \lambda_C

such that :

\lambda_A + \lambda_B + \lambda_C =1,

and :

\overrightarrow=\lambda_A\overrightarrow + \lambda_B\overrightarrow + \lambda_C\overrightarrow,

for every point (for the definition of this arrow notation and further details, see Affine space). For Ceva's theorem, the point is supposed to not belong to any line passing through two vertices of the triangle. This implies that

\lambda_A \lambda_B \lambda_C\ne 0.

If one takes for the intersection of the lines and (see figures), the last equation may be rearranged into :

\overrightarrow-\lambda_C\overrightarrow=\lambda_A\overrightarrow + \lambda_B\overrightarrow.

The left-hand side of this equation is a vector that has the same direction as the line , and the right-hand side has the same direction as the line . These lines have different directions since are not collinear. It follows that the two members of the equation equal the zero vector, and :

\lambda_A\overrightarrow + \lambda_B\overrightarrow=0.

It follows that :

\frac=\frac,

where the left-hand-side fraction is the signed ratio of the lengths of the collinear

s and . The same reasoning shows :

\frac=\frac\quad \text\quad \frac=\frac.

Ceva's theorem results immediately by taking the product of the three last equations.

Generalizations

The theorem can be generalized to higher-dimensional

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

es using barycentric coordinates. Define a cevian of an -simplex as a ray from each vertex to a point on the opposite ()-face ( facet). Then the cevians are concurrent if and only if a

mass distribution In physics and mechanics, mass distribution is the spatial distribution of mass within a solid body. In principle, it is relevant also for gases or liquids, but on Earth their mass distribution is almost homogeneous. Astronomy In astronomy mass d ...

can be assigned to the vertices such that each cevian intersects the opposite facet at its

center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...

. Moreover, the intersection point of the cevians is the center of mass of the simplex. Another generalization to higher-dimensional

es 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 In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle ABC points D, E, and F lie on segments BC, CA, and A ...

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

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

s 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.

References

External links

Menelaus and Ceva
at MathPages
Derivations and applications of Ceva's Theorem
at

cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

Trigonometric Form of Ceva's Theorem
at

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
a

an interactive dynamic geometry sketch using the gravity simulator of Cinderella. * {{DEFAULTSORT:Ceva's Theorem Affine geometry Theorems about triangles Articles containing proofs Euclidean plane geometry

Proofs

Using triangle areas

Generalizations

See also

References

Further reading

External links