We prove a lower bound on the number of ordinary conics determined by a finite point set in R 2 . An ordinary conic for S ⊂ R 2 is a conic that is determined by five points of S, and contains no other points of S. Wiseman and Wilson proved the Sylvester-Gallai-type statement that if a finite point set is not contained in a conic, then it determines at least one ordinary conic. We give a simpler proof of this statement, and then combine it with a theorem of Green and Tao to prove our main result: If S is not contained in a conic and has at most c|S| points on a line, then S determines Ω c (|S| 4 ) ordinary conics. We also give constructions, based on the group law on elliptic curves, that show that the exponent in our bound is best possible.
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