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
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
When this page/section has been completed, {{ProofWanted}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
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): $6$
- 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): $6$