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

Also see

• Pascal's Mystic Hexagram, a variant of this
• Brianchon's Theorem

• Pappus's Hexagon Theorem
• Desargues' Theorem

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