  Archimedes knew that the area between a parabola and any chord $AB$ on the
parabola is four thirds of the area of triangle $\Delta ABP$ 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.
