ON TRIANGLES ASSOCIATED WITH A CURVE

Abstract

It is well-known that the area of parabolic region between a parabola and any chord <TEX>$P_1P_2$</TEX> on the parabola is four thirds of the area of triangle <TEX>${\Delta}P_1P_2P$</TEX>. Here we denote by P the point on the parabola where the tangent is parallel to the chord <TEX>$P_1P_2$</TEX>. In the previous works, the first and third authors of the present paper proved that this property is a characteristic one of parabolas. In this paper, with respect to triangles <TEX>${\Delta}P_1P_2PQ$</TEX> where Q is the intersection point of two tangents to X at <TEX>$P_1$</TEX> and <TEX>$P_2$</TEX> we establish some characterization theorems for parabolas.