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In projective geometry, Pascal's theorem (formulated by Blaise Pascal when he was 16 years old) determines that a hexagon inscribed in a conic, the lines that contain the opposite sides intersect in collinear points, ie if the six vertices a hexagon are located on a circle and three pairs of opposite sides intersect three intersection points are colinear. It is a generalization of the theorem of Pappus.

No doubt a theorem fantastic! Mainly, as well as aesthetic appeal, by Fanto is not clear (at least as far as I know). And it is this that motivates my questions.

1) There is some intuitive way to see Pascal's theorem?

What I mean is something in the same spirit of Java Aplet on the Sum of Outer Angles of a Polygon Theorem.

2) Pascal conceived his theorem as a generalization of Pappus theorem? The proof of Pascal gives some clue as to how he got the idea theorem?

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One thing you can observe is that all non-degenerate conics in the (complex) projective plane are equivalent, so you can reduce (the non-degenerate case) to the case of a circle. – Hurkyl Jul 2 '12 at 1:33
Here is an applet you can play with: – Adeel Jul 2 '12 at 19:13
I don’t know Pascal’s proof of the theorem, but here are a few approaches: 1. A simple argument using projective linear isomorphisms as in §3.2 of 2. By Bézout’s theorem (c.f. §6.2.5 in This argument is generalized by Max Noether’s Fundamentalsatz (c.f. Fulton’s introduction to algebraic curves, p. 123). 3. By Cayley-Bacharach theorem (c.f. Griffith and Harris, p. 673). Hope that helps. – Adeel Jul 2 '12 at 19:32

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