Carnot's polygon theorem is a product-of-ratios relation for intersections with the extended sides
of a polygon. In its spatial form, if a plane
meets the extended sides 
, 
, 
, and 
of a skew quadrilateral 
at finite points 
, 
, 
, and 
, respectively, without containing a vertex,
then
where the bars denote signed lengths on consistently oriented
lines (Altshiller-Court 1979, p. 111).
For a planar polygon with vertices 
,
,
..., 
,
set 
.
Suppose that an algebraic curve of polynomial
degree 
avoids the vertices, has none of the extended sides
as a component, and meets each extended side 
at 
finite points 
, ..., 
, counted with multiplicity.
Then
The identity follows by restricting a defining polynomial 
to each extended side. The inner product is 
, and these ratios
telescope around the polygon.
For a triangle, the cases 
and 
give Menelaus' theorem
and Carnot's conic theorem, respectively
(Szilasi 2012).
See also
Algebraic Curve,
Carnot's Conic Theorem,
Carnot's Theorem,
Carnot's
Theorems,
Menelaus' Theorem,
Polygon,
Skew Quadrilateral
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References
Altshiller-Court, N. "Carnot's Theorem." §329 in Modern
Pure Solid Geometry. New York: Chelsea, p. 111, 1979.Carnot,
L. N. M. Géométrie
de position. Paris, France: Duprat, p. 287, 1803.Carnot,
L. N. M. Mémoir sur la relation qui existe entre les distances
respectives de cinq points quelconques pris dans l'espace; suivi d'un Essai sur la
théorie des transversales. Paris, France: Courcier, p. 71, 1806.Casey,
J. A
Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction
to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin, Ireland:
Hodges, Figgis, & Co., p. 160, 1888.Coolidge, J. L. A
Treatise on Algebraic Plane Curves. New York: Dover, p. 190, 1959.Szilasi,
Z. "Two Applications of the Theorem of Carnot." Ann. Math. Inform. 40,
135-144, 2012. https://ami.uni-eszterhazy.hu/uploads/papers/finalpdf/AMI_40_from135to144.pdf.Referenced
on Wolfram|Alpha
Carnot's Polygon Theorem
Cite this as:
Weisstein, Eric W. "Carnot's Polygon Theorem."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CarnotsPolygonTheorem.html
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