**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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Created: Nov 30 1995 ---
Last modified: Thu Nov 30 15:08:04 1995