ON TRIANGLES ASSOCIATED WITH A CURVE

It is well-known that the area of parabolic region between a parabola and any chord P1P2 on the parabola is four thirds of the area of triangle Delta P1P2P. Here we denote by P the point on the parabola where the tangent is parallel to the chord P1P2. 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 Delta P(1)P(2)Q where Q is the intersection point of two tangents to X at P-1 and P-2 we establish some characterization theorems for parabolas.