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| {{other uses|Ceva (disambiguation)}}
| | La majorité des mutuelles présentent en supplément de leur remboursement en pourcentage du tarif conventionné, un remboursement proposé sous forme de forfait annuel en euros. Il est d’ailleurs plus intéressant pour les personnes disposées aux problèmes de vue. De plus, ce forfait présente l’avantage d’être plus clair pour l’[http://Www.Google.com/search?q=assur%C3%A9&btnI=lucky assuré]. En effet, on vous affiche un montant en euros plutôt qu’un pourcentage. Avec le forfait, pas de problème de compréhension.<br><br>[http://Www.Bbc.Co.uk/search/?q=Feel+free Feel free] to surf to my web blog - [http://www.comparateur-demutuelles.fr comparateur mutuelles] |
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| [[File:Ceva's theorem 1.svg|thumb|right|Ceva's theorem, case 1: the three lines are concurrent at a point O inside ABC]]
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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]]
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| '''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'' to meet opposite sides at ''D'', ''E'' and ''F'' respectively. Then, using signed lengths of segments,
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| :<math>\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1.</math>
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| 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.
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| The [[Theorem#Converse|converse]] is also true: If points ''D'', ''E'' and ''F'' are chosen on ''BC'', ''AC'' and ''AB'' respectively so that
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| : <math>\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1,</math>
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| then ''AD'', ''BE'' and ''CF'' are [[concurrent lines|concurrent]]. The converse is often included as part of the theorem.
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| 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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| Associated with the figures are several terms derived from Ceva's name: ''cevian'' (the lines AD, BE, CF are the cevians of O), ''cevian triangle'' (the triangle DEF is the cevian triangle of O); cevian nest, anticevian triangle, Ceva conjugate. (''Ceva'' is pronounced Chay'va; ''cevian'' is pronounced chev'ian.)
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| The theorem is very similar to [[Menelaus' theorem]] in that their equations differ only in sign.
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| ==Proof of the theorem==
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| A standard proof is as follows:<ref>Follows Russel</ref>
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| First, the sign of the [[left-hand side]] is positive since either all three of the ratios are positive, the case where ''O'' is inside the triangle (upper diagram), or one is positive and the other two are negative, the case ''O'' is outside the triangle (lower diagram shows one case).
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| To check the magnitude, note that the area of a triangle of a given height is proportional to its base. So
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| : <math>\frac{|\triangle BOD|}{|\triangle COD|}=\frac{BD}{DC}=\frac{|\triangle BAD|}{|\triangle CAD|}.</math>
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| Therefore,
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| :<math>\frac{BD}{DC}=
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| \frac{|\triangle BAD|-|\triangle BOD|}{|\triangle CAD|-|\triangle COD|}
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| =\frac{|\triangle ABO|}{|\triangle CAO|}.</math>
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| (Replace the minus with a plus if ''A'' and ''O'' are on opposite sides of ''BC''.)
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| Similarly,
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| : <math>\frac{CE}{EA}=\frac{|\triangle BCO|}{|\triangle ABO|},</math>
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| and
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| : <math>\frac{AF}{FB}=\frac{|\triangle CAO|}{|\triangle BCO|}.</math>
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| Multiplying these three equations gives
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| : <math>\left|\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} \right|= 1,</math>
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| as required.
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| The theorem can also be proven easily using Menelaus' theorem.<ref>Follows {{cite book |title=Inductive Plane Geometry|first=George Irving|last=Hopkins|publisher=D.C. Heath & Co.|year=1902|chapter=Art. 986}}</ref> From the transversal ''BOE'' of triangle ''ACF'',
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| : <math>\frac{AB}{BF} \cdot \frac{FO}{OC} \cdot \frac{CE}{EA} = 1</math>
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| and from the transversal ''AOD'' of triangle ''BCF'',
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| : <math>\frac{BA}{AF} \cdot \frac{FO}{OC} \cdot \frac{CD}{DB} = 1.</math>
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| The theorem follows by dividing these two equations.
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| The converse follows as a corollary.<ref>Follows Russel</ref> Let ''D'', ''E'' and ''F'' be given on the lines ''BC'', ''AC'' and ''AB'' so that the equation holds. Let ''AD'' and ''BE'' meet at ''O'' and let ''F''′ be the point where ''CO'' crosses ''AB''. Then by the theorem, the equation also holds for ''D'', ''E'' and ''F''′. Comparing the two,
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| : <math>\frac{AF}{FB} = \frac{AF'}{F'B}</math>
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| But at most one point can cut a segment in a given ratio so ''F''=''F''′.
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| ==Generalizations==
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| The theorem can be generalized to higher dimensional [[simplex]]es using [[Barycentric coordinates (mathematics)|barycentric coordinates]]. Define a cevian of an ''n''-simplex as a ray from each vertex to a point on the opposite (''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. (Landy. See Wernicke for an earlier result.)
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| [[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.
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| The analogue of the theorem for general [[polygon]]s in the plane has been known since the early nineteenth century {{harv|Grünbaum|Shephard|1995|p=266}}. The theorem has also been generalized to triangles on other surfaces of [[constant curvature]] (Masal'tsev 1994).
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| ==See also==
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| *[[Projective geometry]]
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| *[[Median (geometry)]] - an application
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| ==References==
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| {{reflist}}
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| * {{cite book |title=Pure Geometry
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| |first=John Wellesley|last=Russell|publisher=Clarendon Press|year=1905
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| |chapter= Ch. 1 §7 Ceva's Theorem
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| |url=http://books.google.com/books?id=r3ILAAAAYAAJ}}
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| * {{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=harv | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.
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| * {{cite journal | last1 = Hogendijk | first1 = J. B. | year = 1995 | title = Al-Mutaman ibn Hűd, 11the century king of Saragossa and brilliant mathematician | url = | journal = Historia Mathematica | volume = 22 | issue = | pages = 1–18 | doi = 10.1006/hmat.1995.1001 | ref = harv }}
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| * {{cite journal | last1 = Landy | first1 = Steven |date=December 1988 | title = A Generalization of Ceva's Theorem to Higher Dimensions | url = | journal = The American Mathematical Monthly | volume = 95 | issue = 10| pages = 936–939 | doi = 10.2307/2322390 | ref = harv }}
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| * {{cite journal | last1 = Masal'tsev | first1 = L. A. | year = 1994 | title = Incidence theorems in spaces of constant curvature | url = | journal = Journal of Mathematical Sciences | volume = 72 | issue =4 | pages =3201–3206 |doi= 10.1007/BF01249519 | ref = harv }}
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| * {{cite journal | last1 = Wernicke | first1 = Paul |date=November 1927 | title = The Theorems of Ceva and Menelaus and Their Extension | url = | journal = The American Mathematical Monthly | volume = 34 | issue = 9| pages = 468–472 | doi = 10.2307/2300222 | ref = harv }}
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| ==External links==
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| * [http://www.mathpages.com/home/kmath442/kmath442.htm Menelaus and Ceva] at MathPages
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| * [http://www.cut-the-knot.org/Generalization/ceva.shtml Derivations and applications of Ceva's Theorem] at [[cut-the-knot]]
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| * [http://www.cut-the-knot.org/triangle/TrigCeva.shtml Trigonometric Form of Ceva's Theorem] at [[cut-the-knot]]
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| * [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
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| * [http://forumgeom.fau.edu/FG2001volume1/FG200121.pdf Conics Associated with a Cevian Nest, by Clark Kimberling]
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| *'' [http://demonstrations.wolfram.com/CevasTheorem/ Ceva's Theorem]'' by Jay Warendorff, [[Wolfram Demonstrations Project]].
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| * {{MathWorld |title=Ceva's Theorem |urlname=CevasTheorem}}
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| * [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.
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| {{DEFAULTSORT:Ceva's Theorem}}
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| [[Category:Affine geometry]]
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| [[Category:Triangle geometry]]
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| [[Category:Articles containing proofs]]
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| [[Category:Theorems in plane geometry]]
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La majorité des mutuelles présentent en supplément de leur remboursement en pourcentage du tarif conventionné, un remboursement proposé sous forme de forfait annuel en euros. Il est d’ailleurs plus intéressant pour les personnes disposées aux problèmes de vue. De plus, ce forfait présente l’avantage d’être plus clair pour l’assuré. En effet, on vous affiche un montant en euros plutôt qu’un pourcentage. Avec le forfait, pas de problème de compréhension.
Feel free to surf to my web blog - comparateur mutuelles