CENTROID OF TRIANGLES ASSOCIATED WITH A CURVE

Title & Authors
CENTROID OF TRIANGLES ASSOCIATED WITH A CURVE
Kim, Dong-Soo; Kim, Dong Seo;

Abstract
Archimedes showed that the area between a parabola and any chord AB on the parabola is four thirds of the area of triangle $\small{{\Delta}ABP}$, 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;
Language
English
Cited by
1.
Center of Gravity and a Characterization of Parabolas,;;;

Kyungpook mathematical journal, 2015. vol.55. 2, pp.473-484
1.
Center of Gravity and a Characterization of Parabolas, Kyungpook mathematical journal, 2015, 55, 2, 473
2.
CENTROIDS AND SOME CHARACTERIZATIONS OF PARALLELOGRAMS, Communications of the Korean Mathematical Society, 2016, 31, 3, 637
3.
Areas associated with a Strictly Locally Convex Curve, Kyungpook mathematical journal, 2016, 56, 2, 583
References
1.
A. Benyi, P. Szeptycki, and F. Van Vleck, Archimedean properties of parabolas, Amer. Math. Monthly 107 (2000), no. 10, 945-949.

2.
A. Benyi, P. Szeptycki, and F. Van Vleck, A generalized Archimedean property, Real Anal. Exchange 29 (2003/04), no. 2, 881-889.

3.
W. A. Day, Inequalities for areas associated with conics, Amer. Math. Monthly 98 (1991), no. 1, 36-39.

4.
M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Engle- wood Cliffs, NJ, 1976.

5.
D.-S. Kim and S. H. Kang, A characterization of conic sections, Honam Math. J. 33 (2011), no. 3, 335-340.

6.
D.-S. Kim and Y. H. Kim, A characterization of ellipses, Amer. Math. Monthly 114 (2007), no. 1, 66-70.

7.
D.-S. Kim and Y. H. Kim, Some characterizations of spheres and elliptic paraboloids, Linear Algebra Appl. 437 (2012), no. 1, 113-120.

8.
D.-S. Kim and Y. H. Kim, Some characterizations of spheres and elliptic paraboloids II, Linear Algebra Appl. 438 (2013), no. 3, 1356-1364.

9.
D.-S. Kim and Y. H. Kim, On the Archimedean characterization of parabolas, Bull. Korean Math. Soc. 50 (2013), no. 6, 2103-2114.

10.
D.-S. Kim, W. Kim, Y. H. Kim, and D. H. Park, Area of triangles associated with a curve II, Bull. Korean Math. Soc. 52 (2015), no. 1, 275-286.

11.
D.-S. Kim, Y. H. Kim, and S. Park, Center of gravity and a characterization of parabolas, Kyungpook Math. J., To appear; arXiv:1502.00188.

12.
D.-S. Kim, J. H. Park, and Y. H. Kim, Some characterizations of parabolas, Kyungpook Math. J. 53 (2013), no. 1, 99-104.

13.
D.-S. Kim and K.-C. Shim, Area of triangles associated with a curve, Bull. Korean Math. Soc. 51 (2014), no. 3, 901-909.

14.
J. Krawczyk, On areas associated with a curve, Zesz. Nauk. Uniw. Opol. Mat. 29 (1995), 97-101.

15.
B. Richmond and T. Richmond, How to recognize a parabola, Amer. Math. Monthly 116 (2009), no. 10, 910-922.

16.
S. Stein, Archimedes. What did he do besides cry Eureka?, Mathematical Association of America, Washington, DC, 1999.