On triangles associated with a curve

摘要

It is well-known that the area of parabolic region between a parabola and any chord $P_1P_2$ on the parabola is four thirds of the area of triangle $Delta P_1P_2P$. Here we denote by $P$ the point on the parabola where the tangent is parallel to the chord $P_1P_2$. 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 $Delta P_1P_2Q$ where $Q$ is the intersection point of two tangents to $X$ at $P_1$ and $P_2$ we establish some characterization theorems for parabolas.