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# Pascal's Theorem

The dual of Brianchon's theorem (Casey 1888, p. 146), discovered by B. Pascal in 1640 when he was just 16 years old (Leibniz 1640; Wells 1986, p. 69). It states that, given a (not necessarily regular, or even convex) hexagon inscribed in a conic section, the three pairs of the continuations of opposite sides meet on a straight line, called the Pascal line.

In 1847, Möbius (1885) published the following generalization of Pascal's theorem: if all intersection points (except possibly one) of the lines prolonging two opposite sides of a -gon inscribed in a conic section are collinear, then the same is true for the remaining point.

Braikenridge-Maclaurin Construction, Brianchon's Theorem, Cayley-Bacharach Theorem, Conic Section, Duality Principle, Hexagon, Pappus's Hexagon Theorem, Pascal Lines, Steiner Points, Steiner's Theorem

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## References

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. 129-131, 1888.Casey, J. "Pascal's Theorem." §255 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 145, 328-329, and 354, 1893.Cayley, A. Quart J. 9, p. 348.Coxeter, H. S. M. and Greitzer, S. L. "L'hexagramme de Pascal. Un essai pur reconstituer cette découverte." Le Jeune Scientifique (Joliette, Quebec) 2, 70-72, 1963.Coxeter, H. S. M. and Greitzer, S. L. "Pascal's Theorem." §3.8 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 74-76, 1967.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 44, 1928.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.Forder, H. G. Higher Course Geometry. Cambridge, England: Cambridge University Press, p. 13, 1931.Graustein, W. C. Introduction to Higher Geometry. New York: Macmillan, pp. 260-261, 1930.Johnson, R. A. §386 in Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 236-237, 1929.Lachlan, R. "Pascal's Theorem." §181-191 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 113-119, 1893.Leibniz, G. Letter to M. Périer. In Œuvres de B. Pascal, Vol. 5 (Ed. Bossut). p. 459.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, pp. 105-106, 1990.Pappas, T. "The Mystic Hexagram." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 118, 1989.Perfect, H. Topics in Geometry. London: Pergamon, p. 26, 1963.Salmon, G. §267 and "Notes: Pascal's Theorem, Art. 267" in A Treatise on Conic Sections, 6th ed. New York: Chelsea, pp. 245-246 and 379-382, 1960.Spieker, T. Lehrbuch der ebene Geometrie. Potsdam, Germany, 1888.Veronese. "Nuovi Teremi sull' Hexagrammum Mysticum." Real. Accad. dei Lincei. 1877.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 69, 1986.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 173, 1991.

Pascal's Theorem

## Cite this as:

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