Centroid of triangles associated with a curve
Bull. Korean Math. Soc. 2015 Vol. 52, No. 2, 571-579
https://doi.org/10.4134/BKMS.2015.52.2.571
Published online March 31, 2015
Dong-Soo Kim and Dong Seo Kim
Chonnam National University, Chonnam National University
Abstract : Archimedes showed that the area between a parabola and any chord $AB$ on the parabola is four thirds of the area of triangle $\Delta ABP$, where $P$ is the point on the parabola at which the tangent is parallel to the chord $AB$. Recently, this property of parabolas was proved to be a characteristic property of parabolas. With the aid of this characterization of parabolas, using centroid of triangles associated with a curve we present two conditions which are necessary and sufficient for a strictly locally convex curve in the plane to be a parabola.
Keywords : centroid, parabola, triangle, plane curvature, strictly locally convex curve
MSC numbers : 53A04
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