Pappus's Hexagon Theorem

If A, B, and C are three points on one line, D, E, and F are three points on another line, and AE meets BD at X, AF meets CD at Y, and BF meets CE at Z, then the three points X, Y, and Z are collinear. Pappus's hexagon theorem is self-dual.

The Levi graph of the 9_3 configuration corresponding to the theorem is the Pappus graph.

See also

Brianchon's Theorem, Cayley-Bacharach Theorem, Hexagon, Pappus's Centroid Theorem, Pappus Chain, Pappus Configuration, Pappus Graph, Pappus's Harmonic Theorem, Möbius Tetrahedra, Pascal's Theorem

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References

Coxeter, H. S. M. "Self-Dual Configurations and Regular Graphs." Bull. Amer. Math. Soc. 56, 413-455, 1950.Coxeter, H. S. M. The Beauty of Geometry: Twelve Essays. New York: Dover, p. 244, 1999.Coxeter, H. S. M. and Greitzer, S. L. "Pappus's Theorem." §3.5 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 67-70, 1967.Eves, H. "Pappus' Theorem." §6.2.6 in A Survey of Geometry, rev. ed. Boston, MA: Allyn & Bacon, pp. 79 and 250-251, 1965.Johnson, R. A. "Theorem of Pappus." §388 in Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 237-238, 1929.Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 92-94, 1990.Pappas, T. "Pappus' Theorem & the Nine Coin Puzzle." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 163, 1989.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, pp. 168-169, 1991.

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Pappus's Hexagon Theorem

Cite this as:

Weisstein, Eric W. "Pappus's Hexagon Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PappussHexagonTheorem.html

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