AREA OF TRIANGLES ASSOCIATED WITH A STRICTLY LOCALLY CONVEX CURVE

It is well known that the area U of the triangle formed by three tangents to a parabola X is half of the area T of the triangle formed by joining their points of contact. Recently, it was proved that this property is a characteristic one of parabolas. That is, among strictly locally convex C (3) curves in the plane R 2 parabolas are the only ones satisfying the above area property. In this arti- cle, we study strictly locally convex curves in the plane R 2 . As a result, generalizing the above mentioned characterization theorem for parabolas we present some conditions which are necessary and sufficient for a strictly locally convexC (3) curve in the plane to be an open part of a parabola.