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 (
)-gon circumscribed on a conic section meet in one point,
then the same is true for the remaining line.
See also Duality Principle
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References Casey, J. Dublin: Hodges,
Figgis, & Co., pp. 146-147, 1888. 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. Coxeter, H. S. M.
and Greitzer, S. L. "Brianchon's Theorem." §3.9 in Washington, DC: Math. Assoc. Amer., pp. 77-79, 1967. Geometry
C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. "Extensions
of Pascal's and Brianchon's Theorems." Ch. 2 in London: Stacey International,
pp. 8-30, 1974. The
Seven Circles Theorem and Other New Theorems. Graustein, W. C. New York: Macmillan, p. 261, 1930. Introduction
to Higher Geometry. Johnson,
R. A. §387 in
Boston, MA: Houghton Mifflin, p. 237, 1929. Modern
Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Möbius, F. A.
(Ed. R. Baltzer). Leipzig, Germany: S. Hirzel,
pp. 589-595, 1885. Gesammelte
Werke, Vol. 1 Ogilvy, C. S. New York: Dover, p. 110, 1990. Excursions
in Geometry. Smogorzhevskii,
A. S. New York: Blaisdell, pp. 33-34,
Ruler in Geometrical Constructions. Wells, D. London: Penguin,
pp. 20-21, 1991. The
Penguin Dictionary of Curious and Interesting Geometry. Referenced on Wolfram|Alpha Brianchon's Theorem
Cite this as:
Weisstein, Eric W. "Brianchon's Theorem."
From --A Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/BrianchonsTheorem.html