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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 dual to Desargues' Theorem according to the Duality Principle of Projective Geometry.

See also Cayley-Bacharach Theorem, Desargues' Theorem, Duality Principle, Pascal's Theorem, Projective Geometry


Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 73-74, 1967.

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.

© 1996-9 Eric W. Weisstein