Editing Ceva's theorem (section)

{{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>