# There is some intuitive idea of Pascal's 's theorem in Projective Geometry?

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: cut-the-knot.org/Curriculum/Geometry/Pascal.shtml –  Adeel Khan 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 wwwth.itep.ru/~gorod/ps/stud/projgeom/list.html. 2. By Bézout’s theorem (c.f. §6.2.5 in mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf). 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 Khan Jul 2 '12 at 19:32