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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 [[#References|[1]]].

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 }

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Carnot,   "Géométrie de position" , Paris  (1803)</TD></TR></table>