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 Skip Navigation LinksMath Help > Geometry > Polygons and Triangles > Cyclic Quadrilateral > Pascal's Hexagon Theorem

Pascal's Hexagon Theorem

Pascal's Theorem: if a hexagon is inscribed in a conic, then the three points at which the pairs of opposite sides meet lie on a straight line.

Its dual is Brianchon's Theorem, in which the sides (rather than the vertices) of the hexagon are incident to the conic (i.e. the hexagon is circumscribed), and the intersections of major diagonals (rather than the opposite sides) are concurrent.

Pascal's Theorem applies to a hexagon inscribed in any conic, and so it is a generalization of Pappus' Theorem, which considers only the case of a hexagon inscribed in a pair of lines.
 . . . . . . a proof would be nice!

Internet references

Cut-the-knot: Pascal's Theorem  gives a proof and a Java applet that lets you explore it.

Cut-the-knot: Pascal's Theorem, Homogeneous Coordinates

Related pages in this website:

Summary of geometrical theorems

 

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