Editing Ceva's theorem

{{short description|Geometric relation between line segments from a triangle's vertices and their intersection}}
{{other uses|Ceva (disambiguation)}}

[[File:Ceva's theorem 1.svg|thumb|upright=1.1|Ceva's theorem, case 1: the three lines are concurrent at a point {{mvar|O}} inside {{math|△''ABC''}}]]

[[File:Ceva's theorem 2.svg|thumb|upright=1.1|Ceva's theorem, case 2: the three lines are concurrent at a point {{mvar|O}} outside {{math|△''ABC''}}]]

In [[Euclidean geometry]], '''Ceva's theorem''' is a theorem about [[triangle]]s. Given a triangle {{math|△''ABC''}}, let the [[Line (geometry)|lines]] {{mvar|AO, BO, CO}} be drawn from the [[Vertex (geometry)|vertices]] to a common point {{mvar|O}} (not on one of the sides of {{math|△''ABC''}}), to meet opposite sides at {{mvar|D, E, F}} respectively. (The segments {{mvar|{{overline|AD}}, {{overline|BE}}, {{overline|CF}}}} are known as [[cevian]]s.) Then, using [[Line_segment#Directed_line_segment|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 {{mvar|{{overline|XY}}}} is taken to be positive or negative according to whether {{mvar|X}} is to the left or right of {{mvar|Y}} in some fixed orientation of the line. For example, {{mvar|{{overline|AF}} / {{overline|FB}}}} is defined as having positive value when {{mvar|F}} is between {{mvar|A}} and {{mvar|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 segment]]s that are [[collinearity|collinear]]). It is therefore true for triangles in any [[affine plane]] over any [[field (mathematics)|field]].

A slightly adapted [[Theorem#Converse|converse]] is also true: If points {{mvar|D, E, F}} are chosen on {{mvar|BC, AC, AB}} respectively so that 
: <math>\frac{\overline{AF}}{\overline{FB}}  \cdot \frac{\overline{BD}}{\overline{DC}} \cdot \frac{\overline{CE}}{\overline{EA}} = 1,</math>
then {{mvar|AD, BE, CF}} are [[concurrent lines|concurrent]], or all three [[parallel (geometry)|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 Hud|Yusuf Al-Mu'taman ibn Hűd]], an eleventh-century king of [[Zaragoza]].<ref>{{cite book |title=Geometry: Our Cultural Heritage|url=https://archive.org/details/geometryourcultu00ahol|url-access=limited|first=Audun|last=Holme|publisher=Springer|year=2010|isbn=978-3-642-14440-0|page=[https://archive.org/details/geometryourcultu00ahol/page/n228 210]}}</ref>

Associated with the figures are several terms derived from Ceva's name: [[cevian]] (the lines {{mvar|AD, BE, CF}} are the cevians of {{mvar|O}}), '''cevian triangle''' (the triangle {{math|△''DEF''}} is the cevian triangle of {{mvar|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. By re-writing each in terms of [[Cross-ratio|cross-ratios]], the two theorems may be seen as [[Duality (projective geometry)|projective duals]].<ref>{{Cite journal |last=Benitez |first=Julio |date=2007 |title=A Unified Proof of Ceva and Menelaus' Theorems Using Projective Geometry |url=https://www.heldermann-verlag.de/jgg/jgg11/j11h1beni.pdf |journal=Journal for Geometry and Graphics |volume=11 |issue=1 |pages=39–44}}</ref>

==Proofs==

Several proofs of the theorem have been created.<ref name=r1>{{cite book |title=Pure Geometry
|first=John Wellesley|last=Russell|publisher=Clarendon Press|year=1905
|chapter= Ch. 1 §7 Ceva's Theorem
|url=https://books.google.com/books?id=r3ILAAAAYAAJ}}</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 {{mvar|O}}. 

The second proof uses [[Affine space#Barycentric coordinates|barycentric coordinates]] and [[vector (geometry)|vectors]], but is {{Vague|text=somehow|date=December 2024}} more natural and not case dependent. Moreover, it works in any [[affine plane]] over any [[field (mathematics)|field]].

===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 {{mvar|O}} is inside the triangle (upper diagram), or one is positive and the other two are negative, the case {{mvar|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 
: <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 {{mvar|A}} and {{mvar|O}} are on opposite sides of {{mvar|BC}}.)
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 {{cite book |title=Inductive Plane Geometry|url=https://archive.org/details/inductiveplanege00hopkrich|first=George Irving|last=Hopkins|publisher=D.C. Heath & Co.|year=1902|chapter=Art. 986}}</ref> From the transversal {{mvar|BOE}} of triangle {{math|△''ACF''}},
: <math>\frac{\overline{AB}}{\overline{BF}} \cdot \frac{\overline{FO}}{\overline{OC}} \cdot \frac{\overline{CE}}{\overline{EA}} = -1</math>
and from the transversal {{mvar|AOD}} of triangle {{math|△''BCF''}},
: <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 {{mvar|D, E, F}} be given on the lines {{mvar|BC, AC, AB}} so that the equation holds. Let {{mvar|AD, BE}} meet at {{mvar|O}} and let {{mvar|F'}} be the point where {{mvar|CO}} crosses {{mvar|AB}}. Then by the theorem, the equation also holds for {{mvar|D, E, F'}}. 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 {{mvar|1=F = F’}}.

===Using barycentric coordinates===

Given three points {{mvar|A, B, C}} that are not [[collinearity|collinear]], and a point {{mvar|O}}, that belongs to the same [[plane (geometry)|plane]], the [[Affine space#Barycentric coordinates|barycentric coordinates]] of {{mvar|O}} with respect of {{mvar|A, B, C}} 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 {{mvar|X}} (for the definition of this arrow notation and further details, see [[Affine space]]).

For Ceva's theorem, the point {{mvar|O}} 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 {{mvar|X}} the intersection {{mvar|F}} of the lines {{mvar|AB}} and {{mvar|OC}} (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 {{mvar|CF}}, and the right-hand side has the same direction as the line {{mvar|AB}}. These lines have different directions since {{mvar|A, B, C}} 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 segment]]s {{mvar|{{overline|AF}}}} and {{mvar|{{overline|FB}}}}.

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.

==Generalizations==
The theorem can be generalized to higher-dimensional [[simplex]]es using [[Barycentric coordinates (mathematics)|barycentric coordinates]]. Define a cevian of an {{mvar|n}}-simplex as a ray from each vertex to a point on the opposite ({{math|''n'' − 1}})-face ([[Facet (mathematics)|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>{{cite journal | last1 = Landy | first1 = Steven |date=December 1988 | title = A Generalization of Ceva's Theorem to Higher Dimensions |  journal = The American Mathematical Monthly | volume = 95 | issue = 10| pages = 936–939 | doi = 10.2307/2322390 | jstor = 2322390 }}</ref><ref>{{cite journal | last1 = Wernicke | first1 = Paul |date=November 1927 | title = The Theorems of Ceva and Menelaus and Their Extension |  journal = The American Mathematical Monthly | volume = 34 | issue = 9| pages = 468–472 | doi = 10.2307/2300222| jstor = 2300222 }}</ref>

Another generalization to higher-dimensional [[simplex]]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 {{mvar|k}}-face. This point is the foot of a cevian that goes from the vertex opposite the {{mvar|k}}-face, in a ({{math|''k'' + 1}})-face that contains it, through the point already defined on this ({{math|''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.<ref>{{cite journal | last1 = Samet | first1 = Dov|date=May 2021 | title = An Extension of Ceva's Theorem to ''n''-Simplices |  journal = The American Mathematical Monthly | volume = 128 | issue = 5| pages = 435–445 | doi = 10.1080/00029890.2021.1896292| s2cid = 233413469}}</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 [[polygon]]s in the plane has been known since the early nineteenth century.<ref>{{Cite journal | doi=10.2307/2690569 | last1=Grünbaum | first1=Branko | last2=Shephard | first2=G. C. | title=Ceva, Menelaus and the Area Principle | year=1995 | journal=Mathematics Magazine | volume=68 | issue=4 | pages=254–268 | jstor=2690569 }}</ref>
The theorem has also been generalized to triangles on other surfaces of [[constant curvature]].<ref>{{cite journal | last1 = Masal'tsev | first1 = L. A. | year = 1994 | title = Incidence theorems in spaces of constant curvature |  journal = Journal of Mathematical Sciences | volume = 72 | issue =4 | pages =3201–3206 |doi= 10.1007/BF01249519 | s2cid = 123870381 }}</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.

==See also==
*[[Projective geometry]]
*[[Median (geometry)]] – an application
*[[Circumcevian triangle]]
*[[Menelaus's theorem]]
*[[Triangle]]
*[[Stewart's theorem]]
*[[Cevian]]

==References==
{{reflist}}

==Further reading==
* {{cite journal | last1 = Hogendijk | first1 = J. B. | year = 1995 | title = Al-Mutaman ibn Hűd, 11the century king of Saragossa and brilliant mathematician |  journal = Historia Mathematica | volume = 22 | pages = 1–18 | doi = 10.1006/hmat.1995.1001 | doi-access = free }}

==External links==
* [https://web.archive.org/web/20180809152726/http://www.mathpages.com/home/kmath442/kmath442.htm Menelaus and Ceva] at MathPages
* [http://www.cut-the-knot.org/Generalization/ceva.shtml Derivations and applications of Ceva's Theorem] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/triangle/TrigCeva.shtml Trigonometric Form of Ceva's Theorem] at [[cut-the-knot]]
* [https://web.archive.org/web/20120423103438/http://faculty.evansville.edu/ck6/encyclopedia/glossary.html Glossary of Encyclopedia of Triangle Centers] includes definitions of cevian triangle, cevian nest, anticevian triangle, Ceva conjugate, and cevapoint
* [http://forumgeom.fau.edu/FG2001volume1/FG200121.pdf Conics Associated with a Cevian Nest, by Clark Kimberling]
*'' [http://demonstrations.wolfram.com/CevasTheorem/ Ceva's Theorem]'' by Jay Warendorff, [[Wolfram Demonstrations Project]].
* {{MathWorld |title=Ceva's Theorem |urlname=CevasTheorem}}
* [http://dynamicmathematicslearning.com/finding-centroid-ceva.html Experimentally finding the centroid of a triangle with different weights at the vertices: a practical application of Ceva's theorem] at [http://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches], an interactive dynamic geometry sketch using the gravity simulator of Cinderella.
* {{springer|title=Ceva theorem|id=p/c021370}}

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[[Category:Affine geometry]]
[[Category:Theorems about triangles]]
[[Category:Articles containing proofs]]
[[Category:Euclidean plane geometry]]