Pascal's Theorem

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 as Pascal's mystic hexagram theorem, but that is strictly speaking a specific instance of this more general theorem.


Also see


Source of Name

This entry was named for Blaise Pascal.


Historical Note

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
  • This article incorporates material from proof of Pascal's mystic hexagram on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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