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A theorem on the relation between the lengths of certain lines intersecting a triangle. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c0213701.png" /> be three points lying, respectively, on the sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c0213702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c0213703.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c0213704.png" /> of a triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c0213705.png" />. For the lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c0213706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c0213707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c0213708.png" /> to intersect in a single point or to be all parallel it is necessary and sufficient that
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A theorem on the relation between the lengths of certain lines intersecting a triangle. Let $A_1,B_1,C_1$ be three points lying, respectively, on the sides $BC$, $CA$ and $AB$ of a triangle $ABC$. For the lines $AA_1$, $BB_1$ and $CC_1$ to intersect in a single point or to be all parallel it is necessary and sufficient that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c0213709.png" /></td> </tr></table>
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$$\frac{AC_1}{C_1B}\cdot\frac{BA_1}{A_1C}\cdot\frac{CB_1}{B_1A}=1.$$
  
Lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c02137010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c02137011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c02137012.png" /> that meet in a single point and pass through the vertices of a triangle are called Ceva, or Cevian, lines. Ceva's theorem is metrically dual to the [[Menelaus theorem|Menelaus theorem]]. It is named after G. Ceva [[#References|[1]]].
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Lines $AA_1$, $BB_1$ and $CC_1$ that meet in a single point and pass through the vertices of a triangle are called Ceva, or Cevian, lines. Ceva's theorem is metrically dual to the [[Menelaus theorem|Menelaus theorem]]. It is named after G. Ceva [[#References|[1]]].
  
Ceva's theorem can be generalized to the case of a polygon. Let a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c02137013.png" /> be given in a planar polygon with an odd number of vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c02137014.png" />, and suppose that the lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c02137015.png" /> intersect the sides of the polygon opposite to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c02137016.png" /> respectively in points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c02137017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c02137018.png" />. In this case
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Ceva's theorem can be generalized to the case of a polygon. Let a point $0$ be given in a planar polygon with an odd number of vertices $A_1\dots A_{2n-1}$, and suppose that the lines $0A_1,\dots,0A_{2n-1}$ intersect the sides of the polygon opposite to $A_1,\dots,A_{2n-1}$ respectively in points $a_n,\dots,a_{2n-1}$, $a_1,\dots,a_{n-1}$. In this case
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021370/c02137019.png" /></td> </tr></table>
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$$\frac{A_1a_1}{a_1A_2}\cdot\frac{A_2a_2}{a_2A_3}\cdots\frac{A_{2n-2}a_{2n-2}}{a_{2n-2}A_{2n-1}}\cdot\frac{A_{2n-1}a_{2n-1}}{a_{2n-1}A_1}=1.$$
  
 
====References====
 
====References====

Revision as of 14:35, 19 August 2014

A theorem on the relation between the lengths of certain lines intersecting a triangle. Let $A_1,B_1,C_1$ be three points lying, respectively, on the sides $BC$, $CA$ and $AB$ of a triangle $ABC$. For the lines $AA_1$, $BB_1$ and $CC_1$ to intersect in a single point or to be all parallel it is necessary and sufficient that

$$\frac{AC_1}{C_1B}\cdot\frac{BA_1}{A_1C}\cdot\frac{CB_1}{B_1A}=1.$$

Lines $AA_1$, $BB_1$ and $CC_1$ that meet in a single point and pass through the vertices of a triangle are called Ceva, or Cevian, lines. Ceva's theorem is metrically dual to the Menelaus theorem. It is named after G. Ceva [1].

Ceva's theorem can be generalized to the case of a polygon. Let a point $0$ be given in a planar polygon with an odd number of vertices $A_1\dots A_{2n-1}$, and suppose that the lines $0A_1,\dots,0A_{2n-1}$ intersect the sides of the polygon opposite to $A_1,\dots,A_{2n-1}$ respectively in points $a_n,\dots,a_{2n-1}$, $a_1,\dots,a_{n-1}$. In this case

$$\frac{A_1a_1}{a_1A_2}\cdot\frac{A_2a_2}{a_2A_3}\cdots\frac{A_{2n-2}a_{2n-2}}{a_{2n-2}A_{2n-1}}\cdot\frac{A_{2n-1}a_{2n-1}}{a_{2n-1}A_1}=1.$$

References

[1] G. Ceva, "De lineis rectis se invicem secantibus statica constructio" , Milano (1678)


Comments

References

[a1] M. Berger, "Geometry" , I , Springer (1987)
How to Cite This Entry:
Ceva theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ceva_theorem&oldid=33015
This article was adapted from an original article by P.S. Modenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article