Pascal's Theorem

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Theorem

Let $ABCDEF$ be a hexagon whose $6$ vertices lie on a conic section and whose opposite sides are not parallel.

Then the points of intersection of the opposite sides, when produced as necessary, all lie on a straight line.


Proof

PascalsTheorem.png
PascalsTheorem2.png


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Also known as

Some sources give this theorem as Pascal's mystic hexagram theorem.


Also see


Source of Name

This entry was named for Blaise Pascal.


Historical Note

Pascal's Theorem was discovered by Blaise Pascal when he was in his mid-teens, in the wake of his encounter with Euclid's The Elements.

He published it in his Essay pour les Coniques, which contains $400$ or so corollaries deduced from it, formed by allowing pairs of the six points involved to merge into coincidence.

James Joseph Sylvester called this theorem:

a sort of cat's cradle.


Sources

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