Generalizations to Brianchon's theorem?

Question

I'm wondering if people know of a generalization of Brianchon's theorem. Is there a studied criterion to have a conic (or ellipse) inscribed in a polygon with more than six sides?

Answer 1

My understanding is limited, but you can look at The Heptagon Theorem. The question for that link shows that diagonals of a heptagon with an inscribed conic meet at seven co-conical points. And because the theorem is self dual, I believe this works in reverse.

The theorem comes from Evelyn et al, The Seven Circles Theorem and Other New Theorems. That book also talks about the case of $2n$-sided polygons with inscribed conics, and how the diagonals meet on curves of order $n-2$.

So there are generalizations of Brianchon's theorem to polygons with more than six sides, but involve higher order curves rather than single points.

For another take on this see Katz, Curves In Cages: An Algebro-geometric Zoo, especially Theorem 3.3 (The Mystic 2d-gram). The latter is expressed as a generalization of Pascal's Theorem, but it should dualize to a generalization of Brianchon's Theorem.