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→top: corrected to a more accurate statement: the point O cannot be on a side of ABC (the formula wouldn't be defined), and the converse should admit the "concurrent at infinity" case |
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[[File:Ceva's theorem 2.svg|thumb|250 px|right|Ceva's theorem, case 2: the three lines are concurrent at a point O outside ABC]]
'''Ceva's theorem''' is a theorem about [[triangle]]s in [[Euclidean plane geometry]]. Given a triangle ''ABC'', let the lines ''AO'', ''BO'' and ''CO'' be drawn from the vertices to a common point ''O'' (not on one of the sides of ''ABC''), to meet opposite sides at ''D'', ''E'' and ''F'' respectively. (The segments ''AD, BE,'' and ''CF'' are known as [[cevian]]s.) Then, using signed lengths of segments,
:<math>\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1.</math>
In other words the length ''AB'' is taken to be positive or negative according to whether ''A'' is to the left or right of ''B'' in some fixed orientation of the line. For example, ''AF''/''FB'' is defined as having positive value when ''F'' is between ''A'' and ''B'' and negative otherwise.
: <math>\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1,</math>
then ''AD'', ''BE'' and ''CF'' are [[concurrent lines|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 Hud|Yusuf Al-Mu'taman ibn Hűd]], an eleventh-century king of [[Zaragoza]].<ref>{{cite book |title=Geometry: Our Cultural Heritage|first=Audun|last=Holme|publisher=Springer|year=2010|isbn=3-642-14440-3|page=210}}</ref>
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