On the Archimedean characterization of parabolas

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Archimedes knew that the area between a parabola and any chord AB on the parabola is four thirds of the area of triangle ΔABP where P is the point on the parabola at which the tangent is parallel to AB. We consider whether this property (and similar ones) characterizes parabolas. We present five conditions which are necessary and sufficient for a strictly convex curve in the plane to be a parabola.