# Carnot theorem

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 .

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

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

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