ON THE ARCHIMEDEAN CHARACTERIZATION OF PARABOLAS

Title & Authors
ON THE ARCHIMEDEAN CHARACTERIZATION OF PARABOLAS
Kim, Dong-Soo; Kim, Young Ho;

Abstract
Archimedes knew 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 AB. We consider whether this property (and similar ones) characterizes parabolas. We present five conditions which are necessary and sufficient for a strictly convex curve in the plane to be a parabola.
Keywords
Archimedes;area;parabola;strictly convex curve;curvature;
Language
English
Cited by
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AREA OF TRIANGLES ASSOCIATED WITH A CURVE,;;

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Center of Gravity and a Characterization of Parabolas, Kyungpook mathematical journal, 2015, 55, 2, 473
2.
AREA OF TRIANGLES ASSOCIATED WITH A CURVE, Bulletin of the Korean Mathematical Society, 2014, 51, 3, 901
3.
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4.
CENTROID OF TRIANGLES ASSOCIATED WITH A CURVE, Bulletin of the Korean Mathematical Society, 2015, 52, 2, 571
5.
ON TRIANGLES ASSOCIATED WITH A CURVE, Bulletin of the Korean Mathematical Society, 2015, 52, 3, 925
6.
Areas associated with a Strictly Locally Convex Curve, Kyungpook mathematical journal, 2016, 56, 2, 583
7.
AREA OF TRIANGLES ASSOCIATED WITH A CURVE II, Bulletin of the Korean Mathematical Society, 2015, 52, 1, 275
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AREA OF TRIANGLES ASSOCIATED WITH A STRICTLY LOCALLY CONVEX CURVE, Honam Mathematical Journal, 2015, 37, 1, 41
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Area properties associated with a convex plane curve, Georgian Mathematical Journal, 2017, 24, 3
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.
M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, NJ, 1976.

4.
D.-S. Kim and Y. H. Kim, Some characterizations of spheres and elliptic paraboloids, Linear Algebra Appl. 437 (2012), no. 1, 113-120.

5.
D.-S. Kim and Y. H. Kim, Some characterizations of spheres and elliptic paraboloids. II, Linear Algebra Appl. 438 (2013), no. 3, 1356-1364.

6.
K. Nomizu and T. Sasaki, Affine Differential Geometry, Geometry of affine immersions, Cambridge Tracts in Mathematics, 111, Cambridge University Press, Cambridge, 1994.

7.
B. Richmond and T. Richmond, How to recognize a parabola, Amer. Math. Monthly 116 (2009), no. 10, 910-922.

8.
S. Stein, Archimedes: What did he do besides cry Eureka?, Mathematical Association of America, Washington, DC, 1999.