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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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A slightly adapted [[Theorem#Converse|converse]] is also true: If points ''D'', ''E'' and ''F'' are chosen on ''BC'', ''AC'' and ''AB'' respectively so that
: <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 (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|first=Audun|last=Holme|publisher=Springer|year=2010|isbn=3-642-14440-3|page=210}}</ref>
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