Publisher : Department of Mathematics, Kyungpook National University
DOI : 10.5666/KMJ.2016.56.2.583
Title & Authors
Areas associated with a Strictly Locally Convex Curve Kim, Dong-Soo; Kim, Dong Seo; Kim, Young Ho; Bae, Hyun Seon;
Archimedes showed 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 S of the region bounded by the parabola X and chord AB is four thirds of the area T of triangle . It is well known that the area U formed by three tangents to a parabola is half of the area T of the triangle formed by joining their points of contact. Recently, the first and third authors of the present paper and others proved that among strictly locally convex curves in the plane , these two properties are characteristic ones of parabolas. In this article, in order to generalize the above mentioned property for parabolas we study strictly locally convex curves in the plane satisfying , where and are some functions on the curves. As a result, we present two conditions which are necessary and sufficient for a strictly locally convex curve in the plane to be an open arc of a parabola.