Centroid of triangles associated with a curve

TL;DR

This paper characterizes parabolas among convex curves using centroid properties of triangles associated with the curve, providing necessary and sufficient conditions based on geometric centroid criteria.

Contribution

It introduces new centroid-based conditions that uniquely identify parabolas among strictly locally convex curves.

Findings

Centroid properties characterize parabolas uniquely.

Two necessary and sufficient conditions for parabola identification.

Extension of classical parabola properties to a broader geometric context.

Abstract

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.