CENTROID OF TRIANGLES ASSOCIATED WITH A CURVE
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Archimedes showed that the area between a parabola and any chord AB on the parabola is four thirds of the area of triangle ∆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.
Abstract
Archimedes showed that the area between a parabola and any chord AB on the parabola is four thirds of the area of triangle ∆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.
참고문헌 (16)
- ´A. B´enyi, "Archimedean properties of parabolas", Amer. Math. Monthly, 107(10), pp.945~949 (2000)
- A. B´enyi, "A generalized Archimedean property", Real Anal. Exchange, 29(2), pp.881~889 (2003)
- W. A. Day, "Inequalities for areas associated with conics", Amer. Math. Monthly, 98(1), pp.36~39 (1991)
- M. P. do Carmo, "Differential Geometry of Curves and Surfaces" (1976), Prentice-Hall
- 김동수, "A CHARACTERIZATION OF CONIC SECTIONS", 호남수학학술지, 33(3), pp.335~340 (2011), 호남수학회
- D.-S. Kim, "A characterization of ellipses", Amer. Math. Monthly, 114(1), pp.66~70 (2007)
- D.-S. Kim, "Some characterizations of spheres and elliptic paraboloids", Linear Algebra Appl, 437(1), pp.113~120 (2012)
- D.-S. Kim, "Some characterizations of spheres and elliptic paraboloids II", Linear Algebra Appl, 438(3), pp.1356~1364 (2013)
- 김동수, "On the Archimedean characterization of parabolas", 대한수학회보, 50(6), pp.2103~2114 (2013), 대한수학회
- 김동수, "Area of triangles associated with a curve II", 대한수학회보, 52(1), pp.275~286 (2015), 대한수학회
- 김동수, "Center of Gravity and a Characterization of Parabolas", Kyungpook Mathematical Journal, 55(2), pp.473~484 (2015), 경북대학교 자연과학대학 수학과
- 김동수, "Some Characterizations of Parabolas", Kyungpook Mathematical Journal, 53(1), pp.99~104 (2013), 경북대학교 자연과학대학 수학과
- 김동수, "Area of triangles associated with a curve", 대한수학회보, 51(3), pp.901~909 (2014), 대한수학회
- J. Krawczyk, "On areas associated with a curve", Zesz. Nauk. Uniw. Opol. Mat, 29, pp.97~101 (1995)
- B. Richmond, "How to recognize a parabola", Amer. Math. Monthly, 116(10), pp.910~922 (2009)
- S. Stein, "Archimedes. What did he do besides cry Eureka?" (1999), Mathematical Association of America