Center of gravity and a characterization of parabolas

TL;DR

This paper characterizes parabolas among strictly locally convex plane curves by their unique center of gravity properties, extending Archimedes' classical results to a broader class of curves.

Contribution

It proves that the specific center of gravity properties uniquely characterize parabolas among strictly locally convex plane curves.

Findings

Parabolas are uniquely characterized by their center of gravity properties.

The center of gravity of a parabolic section lies on the axis with a specific ratio.

The properties extend Archimedes' classical results to a broader class of curves.

Abstract

Archimedes determined the center of gravity of a parabolic section as follows. For a parabolic section between a parabola and any chord $AB$ on the parabola, let us denote by $P$ the point on the parabola where the tangent is parallel to $AB$ and by $V$ the point where the line through $P$ parallel to the axis of the parabola meets the chord $AB$. Then the center $G$ of gravity of the section lies on $PV$ called the axis of the parabolic section with $PG=\frac{3}{5}PV$. In this paper, we study strictly locally convex plane curves satisfying the above center of gravity properties. As a result, we prove that among strictly locally convex plane curves, those properties characterize parabolas.