The alpha-version of the Stewart's theorem

• Autorzy: Gelişgen Ö. , Kaya R.
• Języki publikacji: EN

Abstrakty

EN

G. Chen [1] developed Chinese checker metric for the plane on the question “how to develop a metric which would be similar to the movement made by playing Chinese checker” by E. F. Krause [13]. Tian [17] developed α-metric which is defined by dα(P1,P2)=max {|x1 - x2|, |y1 - y2|} + (sec α - tanα) min {|x1 - x2|, |y1 - y2|} where P1= (x1, y1) and P2= (x2, y2) are two points in analytical plane, and α ϵ [0, π/4]. Stewart’s theorem yields a relation between lengths of the sides of a triangle and the length of a cevian of the triangle. A taxicab and Chinese checkers analogues of Stewart’s theorem are given in [12] and [9], respectively. In this work, we give an α-analog of the theorem of Stewart by using the base line concept and we give α-analog of formulae for the medians which is the application of Stewart’s theorem.

Słowa kluczowe

EN

alpha distance; alpha plane geometry; Stewart's theorem; median property

PL

odległość alfa; geometria płaszczyzny alfa; twierdzenie Stewarta; właściwości mediany

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2013
• Tom: Vol. 46, nr 4
• Strony: 795--808
• Opis fizyczny: Bibliogr. 17 poz., rys., tab., wykr.

Twórcy

autor

Gelişgen Ö.

• Department of Mathematics and Computer Sciences, Faculty of Science and Arts, University of Eskisehir Osmangazi, Eskisehir, Turkey

autor

Kaya R.

• Department of Mathematics and Computer Sciences, Faculty of Science and Arts, University of Eskisehir Osmangazi, Eskisehir, Turkey

Bibliografia

• [1] G. Chen, Lines and Circles in Taxicab Geometry, Master Thesis, Department of Mathematics and Computer Science, Central Missouri State Univ., 1992.
• [2] H. S. M. Coxeter, S. L. Greitzer, Geometry Revisited, The Mathematical Association of America, 1967.
• [3] H. B. Çolakoğlu, Ö. Gelişgen, R. Kaya, Pythagorean theorems in the alpha plane, Math. Commun. 14(2) (2009), 211–221.
• [4] B. Divjak, Notes on taxicab geometry, Scientific and Professional Information Journal of Croatian Society for Constructive Geometry and Computer Graphics (KoG) 5 (2000), 5–9.
• [5] Ö. Gelisgen, Minkowski Geometrileri Üzerine: Taksi, Çin Dama ve-Geometrileri Hakkında Genel Bir Analiz, Phd Thesis, Eskişehir Osmangazi University, 2007.
• [6] Ö. Gelişgen, R. Kaya, CC-analog of the theorem of Pythagoras, Algebras Groups Geom. 23(2) (2006), 179–188.
• [7] Ö. Gelişgen, R. Kaya, On distance in three dimensional space, Appl. Sci. 8 (2006), 65–69.
• [8] Ö. Gelişgen, R. Kaya, Generalization of distance to n-dimensional space, Scientific and Professional Journal of the Croatian Society for Geometry and Graphics (KoG) 10 (2006), 33–37.
• [9] Ö. Gelişgen, R. Kaya, The CC-version of the Stewart’s theorem, Appl. Sci. 11 (2009), 68–77.
• [10] R. Kaya, Ö. Gelişgen, S. Ekmekçi, A. Bayar, Group of isometries of CC-plane, Missouri J. Math. Sci. 18(3) (2006), 221–233.
• [11] R. Kaya, Ö. Gelişgen, S. Ekmekçi, A. Bayar, On the group of isometries of the plane with generalized absolute value metric, Rocky Mountain J. Math. 39(2) (2009), 591–604.
• [12] R. Kaya, H. B. Colakoglu, Taxicab versions of some Euclidean theorems, Int. J. Pure Appl. Math. 26(1) (2006), 69–81.
• [13] E. F. Krause, Taxicab Geometry, Addision-Wesley, Menlo Park, California, 1975.
• [14] M. Özcan, R. Kaya, Area of a triangle in terms of the taxicab distance, Missouri J. Math. Sci. 15(3) (2003), 178–185.
• [15] R. S. Milmann, G. D. Parker, Geometry; A Metric Approach with Models, Springer, 1991.
• [16] A. C. Thompson, Minkowski Geometry, Cambridge University Press, 1996.
• [17] S. Tian, Alpha distance – a generalization of Chinese checker distance and taxicab distance, Missouri J. Math. Sci. 17(1) (2005), 35–40.