It is given a hexagon inscribed in a conic section.
I want to prove that the pairs of opposite site intersect at three points of the projective plane that are collinear.
How could we do this? Could you give me some hints?
2026-03-29 10:55:52.1774781752
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The intersection points are collinear
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Here is a link for a beautiful paper by Eisenbud, Harris, and Green:
Eisenbud, David; Green, Mark; Harris, Joe: Cayley-Bacharach theorems and conjectures. Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 3, 295–324.
See Theorem CB1, Theorem CB2, and Theorem CB3. The question you asked is actually Theorem CB2.
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Oops I said conic when I meant cubic. Anyway, here is an expansion on my comment. Let the first cubic be the lines defined by the first, third, and fifth sides of the hexagon. Let the second cubic be the lines defined by the remaining sides. These two conics intersect at the 6 vertices of the hexagon and the three points you care about. Now let the third cubic be the conic and a line through two of the points you care about. This contains 8 of the 9 points I mentioned, so by Cayley-Bacharach it must contain the ninth.