# 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

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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

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{V}$: "Greatness and Misery of Man" - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $6$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Pascal's mystic hexagram theorem** - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.16$: Pascal ($\text {1623}$ – $\text {1662}$) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $6$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Pascal's theorem** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Pascal's theorem**