Namespaces
Variants
Actions

Difference between revisions of "Carnot theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A theorem on the product of the simple ratios in which the points of intersection of an algebraic curve with the sides of a triangle divide these sides. Suppose that the algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c0204901.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c0204902.png" /> does not pass through any of the vertices of a triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c0204903.png" /> and intersects each side, extended if necessary, at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c0204904.png" /> points: the side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c0204905.png" /> at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c0204906.png" />; the side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c0204907.png" /> at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c0204908.png" />; and the side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c0204909.png" /> at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049010.png" />. Then the product of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049011.png" /> simple ratios
+
<!--
 +
c0204901.png
 +
$#A+1 = 29 n = 0
 +
$#C+1 = 29 : ~/encyclopedia/old_files/data/C020/C.0200490 Carnot theorem
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/c020/c020490/c02049012.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049013.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049014.png" /> is odd, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049015.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049016.png" /> is even.
+
A theorem on the product of the simple ratios in which the points of intersection of an algebraic curve with the sides of a triangle divide these sides. Suppose that the algebraic curve  $  l $
 +
of order  $  n $
 +
does not pass through any of the vertices of a triangle  $  A B C $
 +
and intersects each side, extended if necessary, at  $  n $
 +
points: the side  $  A B $
 +
at the points  $  C _ {1} \dots C _ {n} $;
 +
the side  $  B C $
 +
at the points  $  A _ {1} \dots A _ {n} $;
 +
and the side  $  C A $
 +
at the points  $  B _ {1} \dots B _ {n} $.  
 +
Then the product of the  $  3 n $
 +
simple ratios
  
This statement is equivalent to the following: The product of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049017.png" /> ratios
+
$$
  
<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/c020/c020490/c02049018.png" /></td> </tr></table>
+
\frac{ {A C _ {i} } ^  \rightarrow  }{ {C _ {i} B } ^  \rightarrow  }
 +
,\ \
  
is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049019.png" />.
+
\frac{ {B A _ {i} } ^  \rightarrow  }{ {A _ {i} C } ^  \rightarrow  }
 +
,\ \
 +
 
 +
\frac{ {C B _ {i} } ^  \rightarrow  }{ {B _ {i} A } ^  \rightarrow  }
 +
,\ \
 +
i = 1 \dots n ,
 +
$$
 +
 
 +
is equal to $  - 1 $
 +
if  $  n $
 +
is odd, and  $  + 1 $
 +
if  $  n $
 +
is even.
 +
 
 +
This statement is equivalent to the following: The product of the  $  3 n $
 +
ratios
 +
 
 +
$$
 +
 
 +
\frac{ {C _ {i} A } ^  \rightarrow  }{ {C _ {i} B } ^  \rightarrow  }
 +
,\ \
 +
 
 +
\frac{ {A _ {i} B } ^  \rightarrow  }{ {A _ {i} C } ^  \rightarrow  }
 +
,\ \
 +
 
 +
\frac{ {B _ {i} C } ^  \rightarrow  }{ {B _ {i} A } ^  \rightarrow  }
 +
,\ \
 +
i = 1 \dots n ,
 +
$$
 +
 
 +
is equal to  $  + 1 $.
  
 
A special case of this theorem was proved by L. Carnot [[#References|[1]]].
 
A special case of this theorem was proved by L. Carnot [[#References|[1]]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049020.png" /> is a straight line then the [[Menelaus theorem|Menelaus theorem]] is obtained. A generalization of Carnot's theorem is: Suppose that an algebraic curve of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049021.png" /> intersects each of the straight lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049024.png" />, lying in the plane of this curve, at exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049025.png" /> points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049027.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020490/c02049028.png" />. Then
+
If $  l $
 +
is a straight line then the [[Menelaus theorem|Menelaus theorem]] is obtained. A generalization of Carnot's theorem is: Suppose that an algebraic curve of order $  n $
 +
intersects each of the straight lines $  A _ {i} A _ {i+1} $,  
 +
$  i = 1 \dots m $,  
 +
$  A _ {m+1} = A _ {1} $,  
 +
lying in the plane of this curve, at exactly $  n $
 +
points $  B _ {ij} $,  
 +
$  i = 1 \dots m $;  
 +
$  j = 1 \dots n $.  
 +
Then
 +
 
 +
$$
 +
\prod _ { i,j }
  
<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/c020/c020490/c02049029.png" /></td> </tr></table>
+
\frac{ {A _ {i} B _ {ij} } ^  \rightarrow  }{ {B _ {ij} A _ {i+1} } ^  \rightarrow  }
 +
\
 +
= ( - 1 )  ^ {mn} .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Carnot,  "Géométrie de position" , Paris  (1803)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Carnot,  "Géométrie de position" , Paris  (1803)</TD></TR></table>

Latest revision as of 10:08, 4 June 2020


A theorem on the product of the simple ratios in which the points of intersection of an algebraic curve with the sides of a triangle divide these sides. Suppose that the algebraic curve $ l $ of order $ n $ does not pass through any of the vertices of a triangle $ A B C $ and intersects each side, extended if necessary, at $ n $ points: the side $ A B $ at the points $ C _ {1} \dots C _ {n} $; the side $ B C $ at the points $ A _ {1} \dots A _ {n} $; and the side $ C A $ at the points $ B _ {1} \dots B _ {n} $. Then the product of the $ 3 n $ simple ratios

$$ \frac{ {A C _ {i} } ^ \rightarrow }{ {C _ {i} B } ^ \rightarrow } ,\ \ \frac{ {B A _ {i} } ^ \rightarrow }{ {A _ {i} C } ^ \rightarrow } ,\ \ \frac{ {C B _ {i} } ^ \rightarrow }{ {B _ {i} A } ^ \rightarrow } ,\ \ i = 1 \dots n , $$

is equal to $ - 1 $ if $ n $ is odd, and $ + 1 $ if $ n $ is even.

This statement is equivalent to the following: The product of the $ 3 n $ ratios

$$ \frac{ {C _ {i} A } ^ \rightarrow }{ {C _ {i} B } ^ \rightarrow } ,\ \ \frac{ {A _ {i} B } ^ \rightarrow }{ {A _ {i} C } ^ \rightarrow } ,\ \ \frac{ {B _ {i} C } ^ \rightarrow }{ {B _ {i} A } ^ \rightarrow } ,\ \ i = 1 \dots n , $$

is equal to $ + 1 $.

A special case of this theorem was proved by L. Carnot [1].

If $ l $ is a straight line then the Menelaus theorem is obtained. A generalization of Carnot's theorem is: Suppose that an algebraic curve of order $ n $ intersects each of the straight lines $ A _ {i} A _ {i+1} $, $ i = 1 \dots m $, $ A _ {m+1} = A _ {1} $, lying in the plane of this curve, at exactly $ n $ points $ B _ {ij} $, $ i = 1 \dots m $; $ j = 1 \dots n $. Then

$$ \prod _ { i,j } \frac{ {A _ {i} B _ {ij} } ^ \rightarrow }{ {B _ {ij} A _ {i+1} } ^ \rightarrow } \ = ( - 1 ) ^ {mn} . $$

References

[1] L. Carnot, "Géométrie de position" , Paris (1803)
How to Cite This Entry:
Carnot theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carnot_theorem&oldid=46259
This article was adapted from an original article by P.S. Modenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article