Ceva's theorem

Template:Short description {{#invoke:other uses|otheruses}}

Ceva's theorem, case 1: the three lines are concurrent at a point Template:Mvar inside Template:Math

Ceva's theorem, case 2: the three lines are concurrent at a point Template:Mvar outside Template:Math

In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle Template:Math, let the lines Template:Mvar be drawn from the vertices to a common point Template:Mvar (not on one of the sides of Template:Math), to meet opposite sides at Template:Mvar respectively. (The segments Template:Mvar 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 Template:Mvar is taken to be positive or negative according to whether Template:Mvar is to the left or right of Template:Mvar in some fixed orientation of the line. For example, Template:Mvar is defined as having positive value when Template:Mvar is between Template:Mvar and Template:Mvar 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 Template:Mvar are chosen on Template:Mvar respectively so that

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

then Template:Mvar 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.<ref>Template:Cite book</ref>

Associated with the figures are several terms derived from Ceva's name: cevian (the lines Template:Mvar are the cevians of Template:Mvar), cevian triangle (the triangle Template:Math is the cevian triangle of Template:Mvar); 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.<ref>Template:Cite journal</ref>

ProofsEdit

Several proofs of the theorem have been created.<ref name=r1>Template:Cite book</ref><ref>Alfred S. Posamentier and Charles T. Salkind (1996), Challenging Problems in Geometry, pages 177–180, Dover Publishing Co., second revised edition.</ref> Two proofs are given in the following.

The first one is very elementary, using only basic properties of triangle areas.<ref name=r1 /> However, several cases have to be considered, depending on the position of the point Template:Mvar.

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

Using triangle areasEdit

First, the sign of the left-hand side is positive since either all three of the ratios are positive, the case where Template:Mvar is inside the triangle (upper diagram), or one is positive and the other two are negative, the case Template:Mvar 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

<math>\frac{|\triangle BOD|}{|\triangle COD|}=\frac{\overline{BD}}{\overline{DC}}=\frac{|\triangle BAD|}{|\triangle CAD|}.</math>

Therefore,

<math>\frac{\overline{BD}}{\overline{DC}}=

\frac{|\triangle BAD|-|\triangle BOD|}{|\triangle CAD|-|\triangle COD|} =\frac{|\triangle ABO|}{|\triangle CAO|}.</math> (Replace the minus with a plus if Template:Mvar and Template:Mvar are on opposite sides of Template:Mvar.) Similarly,

<math>\frac{\overline{CE}}{\overline{EA}}=\frac{|\triangle BCO|}{|\triangle ABO|},</math>

and

<math>\frac{\overline{AF}}{\overline{FB}}=\frac{|\triangle CAO|}{|\triangle BCO|}.</math>

Multiplying these three equations gives

<math>\left|\frac{\overline{AF}}{\overline{FB}} \cdot \frac{\overline{BD}}{\overline{DC}} \cdot \frac{\overline{CE}}{\overline{EA}} \right|= 1,</math>

as required.

The theorem can also be proven easily using Menelaus's theorem.<ref>Follows Template:Cite book</ref> From the transversal Template:Mvar of triangle Template:Math,

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

and from the transversal Template:Mvar of triangle Template:Math,

<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.<ref name=r1/> Let Template:Mvar be given on the lines Template:Mvar so that the equation holds. Let Template:Mvar meet at Template:Mvar and let Template:Mvar be the point where Template:Mvar crosses Template:Mvar. Then by the theorem, the equation also holds for Template:Mvar. Comparing the two,

<math>\frac{\overline{AF}}{\overline{FB}} = \frac{\overline{AF'}}{\overline{F'B}}</math>

But at most one point can cut a segment in a given ratio so Template:Mvar.

Using barycentric coordinatesEdit

Given three points Template:Mvar that are not collinear, and a point Template:Mvar, that belongs to the same plane, the barycentric coordinates of Template:Mvar with respect of Template:Mvar are the unique three numbers <math>\lambda_A, \lambda_B, \lambda_C</math> such that

<math>\lambda_A + \lambda_B + \lambda_C =1,</math>

and

<math>\overrightarrow{XO}=\lambda_A\overrightarrow{XA} + \lambda_B\overrightarrow{XB} + \lambda_C\overrightarrow{XC},</math>

for every point Template:Mvar (for the definition of this arrow notation and further details, see Affine space).

For Ceva's theorem, the point Template:Mvar is supposed to not belong to any line passing through two vertices of the triangle. This implies that <math>\lambda_A \lambda_B \lambda_C\ne 0.</math>

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

<math>\overrightarrow{FO}-\lambda_C\overrightarrow{FC}=\lambda_A\overrightarrow{FA} + \lambda_B\overrightarrow{FB}.</math>

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

<math>\lambda_A\overrightarrow{FA} + \lambda_B\overrightarrow{FB}=0.</math>

It follows that

<math>\frac{\overline{AF}}{\overline{FB}}=\frac{\lambda_B}{\lambda_A},</math>

where the left-hand-side fraction is the signed ratio of the lengths of the collinear line segments Template:Mvar and Template:Mvar.

The same reasoning shows

<math>\frac{\overline{BD}}{\overline{DC}}=\frac{\lambda_C}{\lambda_B}\quad \text{and}\quad \frac{\overline{CE}}{\overline{EA}}=\frac{\lambda_A}{\lambda_C}.</math>

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

GeneralizationsEdit

The theorem can be generalized to higher-dimensional simplexes using barycentric coordinates. Define a cevian of an Template:Mvar-simplex as a ray from each vertex to a point on the opposite (Template:Math)-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.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>

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 Template:Mvar-face. This point is the foot of a cevian that goes from the vertex opposite the Template:Mvar-face, in a (Template:Math)-face that contains it, through the point already defined on this (Template:Math)-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.<ref>Template:Cite journal</ref>

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.<ref>Template:Cite journal</ref> The theorem has also been generalized to triangles on other surfaces of constant curvature.<ref>Template:Cite journal</ref>

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.

ReferencesEdit

Template:Reflist

External linksEdit

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.
{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:CevasTheorem%7CCevasTheorem.html}} |title = Ceva's Theorem |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

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.
Template:Springer