ON TRIANGLES ASSOCIATED WITH A CURVE

. It is well-known that the area of parabolic region between aparabola and any chord P 1 P 2 on the parabola is four thirds of the area oftriangle ∆P 1 P 2 P. Here we denote by P the point on the parabola wherethe tangent is parallel to the chord P 1 P 2 . In the previous works, therst and third authors of the present paper proved that this property isa characteristic one of parabolas. In this paper, with respect to triangles∆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. 1. IntroductionFor a smooth function f : I → Rdened on an open interval, usually we saythat f is strictly convex if the graph of f has positive curvature κ with respectto the upward unit normal N. This condition is equivalent to f ′′ (x) > 0 on theinterval I.In this article, we study strictly locally convex plane curves. A regular planecurve X : I → R 2 dened on an open interval I, is called convex if, for all s ∈ Ithe trace X(I) of X lies entirely on one side of the closed half-plane determinedby the tangent line to X at s ([4]). A regular plane curve X : I → R