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Carnot's Polygon Theorem


If a plane cuts the sides AB, BC, CD, and DA of a skew quadrilateral ABCD in points P, Q, R, and S, then

 (AP)/(PB)·(BQ)/(QC)·(CR)/(RD)·(DS)/(SA)=1

both in magnitude and sign (Altshiller-Court 1979, p. 111).

More generally, if P_1, P_2, ..., are the polygon vertices of a finite polygon with no "minimal sides" and the side P_iP_j meets a curve in the points P_(ij1) and P_(ij2), then

 (product_(i)P_1P_(12i)^_product_(i)P_2P_(23i)^_...product_(i)P_NP_(N1i)^_)/(product_(i)P_NP_(N1i)^_...product_(i)P_2P_(2i1)^_)=1,

where AB^_ denotes the distance from point A to B.


See also

Carnot's Theorem

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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: 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: 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: Hodges, Figgis, & Co., p. 160, 1888.Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 190, 1959.

Referenced on Wolfram|Alpha

Carnot's Polygon Theorem

Cite this as:

Weisstein, Eric W. "Carnot's Polygon Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CarnotsPolygonTheorem.html

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