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# Some Stronger Theorems

We know that Pascal's theorem and Brianchon's theorem are true of conics, but we can also make a stronger claim.

Theorem: Pascal's theorem characterizes conics. In other words, say a curve is given such that opposite sides of any inscribed hexagon intersect in three collinear points. Then the curve is part of a conic.

Proof: Say there are five points or fewer on the curve. Then, because five points determine a conic, all the points on the curve lie on at least one conic.

So say there are more than five points on the curve. Call five of them ABCDE; these five determine a conic. Take any other point F on the curve. Then by hypothesis, ABCDEF has opposite sides which intersect in collinear points. So, by the converse of Pascal's theorem, it lies on a conic; in other words, F lies on the conic determined by ABCDE. So every point on the curve lies on the conic determined by ABCDE, and the curve is part of that conic. QED.

The dual of the above theorem is that Brianchon's theorem characterizes conics, and its proof is the dual of the given proof.

I have never seen these theorems stated before, although they seem too simple to be new (and indeed I have seen some slight variants). If you can find someone else who has stated either of these theorems, please let me know.

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