Archimedes showed 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 the chord $AB$. Recently, this property of parabolas was proved to be a characteristic property of parabolas. With the aid of this characterization of parabolas, using centroid of triangles associated with a curve we present two conditions which are necessary and sufficient for a strictly locally convex curve in the plane to be a parabola.
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机译：阿基米德表明，抛物线与抛物线上任何和弦$ AB $之间的面积是三角形$ Delta ABP $面积的四分之三，其中$ P $是抛物线上切线与弦平行的点$ AB $。最近，抛物线的这种特性被证明是抛物线的特征。借助于抛物线的这种表征，使用与曲线相关的三角形的质心，我们提出了两个条件，这两个条件对于将平面中严格局部凸的曲线作为抛物线是必要的和充分的。
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