It is well known that the area $U$ of the triangleformed by three tangents to a parabola $X$ is half of the area $T$of thetriangle 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${\mathbb R}^{2}$ parabolas are the only ones satisfying the above area property. In this article, we study strictly locally convex curves in the plane${\mathbb R}^{2}$. As a result, generalizing the above mentioned characterization theorem for parabolaswe present some conditions which are necessary and sufficient fora strictly locally convex $C^{(3)}$ curve in the plane to be an open part of a parabola.