On the Archimedean characterization of parabolas

TL;DR

This paper investigates whether specific geometric properties uniquely characterize parabolas, presenting five necessary and sufficient conditions for a convex curve to be a parabola based on Archimedean properties.

Contribution

The paper introduces five new conditions that precisely characterize parabolas among convex curves, extending classical Archimedean properties.

Findings

Five conditions are necessary and sufficient for a convex curve to be a parabola.

Archimedean properties can be used to characterize parabolas uniquely.

The results deepen understanding of parabola geometry and its unique properties.

Abstract

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.