Bulletin of the
Korean Mathematical Society BKMS

Archimedes proved that for a point $P$ on a parabola $X$ and a chord $AB$ of $X$ parallel to the tangent of $X$ at $P$, the area of the region bounded by the parabola $X$ and the chord $AB$ is four thirds of the area of the triangle $\bigtriangleup ABP$. This property was proved to be a characteristic of parabolas, so called the Archimedean characterization of parabolas. In this article, we study strictly convex curves in the plane ${\mathbb R}^{2}$. As a result, first using a functional equation we establish a characterization theorem for quadrics. With the help of this characterization we give another proof of the Archimedean characterization of parabolas. Finally, we present two related conditions which are necessary and sufficient for a strictly convex curve in the plane to be an open arc of a parabola.

Keywords: Triangle, area, parabola, strictly convex curve, plane curvature, quadric, Archimedean
characterization of parabolas

MSC numbers: 53A04

Supported by: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A3B05050223).