Pascal's Theorem
The dual of Brianchon's Theorem. It states that, given a (not necessarily Regular, oreven Convex) Hexagon inscribed in a Conic Section, the three pairs of thecontinuations of opposite sides meet on a straight Line, called the Pascal Line. There are 6! (6! means 6Factorial, where 
) possible ways of taking all Vertices in any order, but among these are six equivalent Cyclic Permutations and twopossible orderings, so the total number of different hexagons (not all simple) is
There are therefore a total of 60 Pascal Lines created by connecting Vertices in any order. These intersect three by three in 20 Steiner Points.See also Braikenridge-Maclaurin Construction, Brianchon's Theorem, Cayley-Bacharach Theorem, ConicSection, Duality Principle, Hexagon, Pappus's Hexagon Theorem, Pascal Line, SteinerPoints
References
Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 73-76, 1967. 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.
- Mean Run Density
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