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 [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)