Brianchon's Theorem


The dual of Pascal's theorem (Casey 1888, p. 146). It states that, given a hexagon circumscribed on a conic section, the lines joining opposite polygon vertices (polygon diagonals) meet in a single point.

In 1847, Möbius (1885) gave a statement which generalizes Brianchon's theorem: if all lines (except possibly one) connecting two opposite vertices of a (4n+2)-gon circumscribed on a conic section meet in one point, then the same is true for the remaining line.

See also

Duality Principle, Pascal's Theorem

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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., pp. 146-147, 1888.Coxeter, H. S. M. and Greitzer, S. L. "Brianchon's Theorem." §3.9 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 77-79, 1967.Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "Extensions of Pascal's and Brianchon's Theorems." Ch. 2 in The Seven Circles Theorem and Other New Theorems. London: Stacey International, pp. 8-30, 1974.Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, p. 261, 1930.Johnson, R. A. §387 in Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 237, 1929.Möbius, F. A. Gesammelte Werke, Vol. 1 (Ed. R. Baltzer). Leipzig, Germany: S. Hirzel, pp. 589-595, 1885.Ogilvy, C. S. Excursions in Geometry. New York: Dover, p. 110, 1990.Smogorzhevskii, A. S. The Ruler in Geometrical Constructions. New York: Blaisdell, pp. 33-34, 1961.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 20-21, 1991.

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Brianchon's Theorem

Cite this as:

Weisstein, Eric W. "Brianchon's Theorem." From MathWorld--A Wolfram Web Resource.

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