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 of order
does not pass through any of the vertices of a triangle
and intersects each side, extended if necessary, at
points: the side
at the points
; the side
at the points
; and the side
at the points
. Then the product of the
simple ratios
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is equal to if
is odd, and
if
is even.
This statement is equivalent to the following: The product of the ratios
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is equal to .
A special case of this theorem was proved by L. Carnot [1].
If is a straight line then the Menelaus theorem is obtained. A generalization of Carnot's theorem is: Suppose that an algebraic curve of order
intersects each of the straight lines
,
,
, lying in the plane of this curve, at exactly
points
,
;
. Then
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References
[1] | L. Carnot, "Géométrie de position" , Paris (1803) |
Carnot theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carnot_theorem&oldid=46259