Euler Line And The Nine-Point Circle

  IJMTT-book-cover
 
International Journal of Mathematics Trends and Technology (IJMTT)
 
© 2021 by IJMTT Journal
Volume-67 Issue-9
Year of Publication : 2021
Authors : Aleksa Srdanov, Dragan Stojiljkovic, Andreja Lazic
  10.14445/22315373/IJMTT-V67I9P508

MLA

MLA Style: Aleksa Srdanov, Dragan Stojiljkovic, Andreja Lazic"Euler Line And The Nine-Point Circle" International Journal of Mathematics Trends and Technology 67.9 (2021):72-80. 

APA Style: Aleksa Srdanov, Dragan Stojiljkovic, Andreja Lazic(2021). Euler Line And The Nine-Point Circle  International Journal of Mathematics Trends and Technology, 67(9), 72-80.

Abstract
We will show how the way of defining mathematical concepts significantly affects the possibility of generalising them. The method of a proof provides an opportunity for a better understanding of the nature of problems and their solutions. The paper will present a slightly different way of understanding the tetrahedron orthogonality concept. Various ways and possibilities of generalisation are also considered.

Reference

[1] Malgorzata Buba-Brzozowa. Analogues of the nine-point circle for orthocentric n-simplexes. J. Geom., 81(1-2) (2004) 21–29.
[2] Malgorzata Buba-Brzozowa. The Monge point and the 3(n + 1) point sphere of an n-simplex. J. Geom. Graph., 9(1) (2005) 31–36.
[3] Javier Alonso, Horst Martini, and Senlin Wu. On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces. Aequationes Math., 83(1-2) (2012) 153– 189.
[4] J. Alonso, H. Martini and S. Wu. On Birkhoff orthogonality and isosceles orthogonality in normedlinear spaces. Aequationes Math.83(2012) 153–189.
[5] E. Asplund and B. Grünbaum. On the geometry of Minkowski planes. L’Enseignement Mathematique 6(2) (1961) 299–306.
[6] R. Crabbs. Gaspard Monge and the Monge point of the tetrahedron. Mathematics Magazine76(3) (2003) 193–203.
[7] A. Edmonds, M. Hajja and H. Martini. Orthocentric simplices and their centers. Results Math.47 (2005) 266–295.
[8] M, Hajja and H. Martini. Orthocentric simplices as the true generalizations of triangles. The Mathematical Intelligencer 35 (2013) 16–27.
[9] H. Martini and M. Spirova. The Feuerbach circle and orthocentricity in normed planes. L’EnseignementMathematique, 53(2) (2007) 237–258.
[10] H. Martini and S, Wu. On orthocentric systems in strictly convex normed planes. Extracta Math.24(2009) 31–45.
[11] W. Pacheco and T. Rosas. On orthocentric systems in Minkowski planes. Beitr. Algebra Geom.56(2015) 249–262.
[12](PDF) Orthocenters of triangles in n-dimensional space. Available from: https://www.researchgate.net/publication/283490962_Orthocenters_of_triangles_in_n-dimensional_space .
[13] П. С. Моденов. Задачи по геометрији. Наука, Москва, (1979).
[14] Aleksa Srdanov. Euler line – generalisation. Зборник радова ВТШ Пожаревац 1/2011.
[15] Aleksa Srdanov. Nine-point circle – dimensional generalisation. Зборник радова ВТШ Пожаревац 1/2012.
[16] Aleksa Srdanov. Feuerbach circle – dimensional generalisation. Зборник радова ВТШ Пожаревац 1/2012.
[17] И. Х. Сивашинскии. Пособие по математике дла техникумов. Виша школа Москва. (зад.395), (1970).
[18] Honsberger, R. The Nine-Point Circle. §1.3 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., (1995) 6-7.
[19] Feuerbach, Karl Wilhelm; Buzengeiger, Carl Heribert Ignatz . Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten Linien und Figuren. Eine analytisch-trigonometrische Abhandlung (Monograph ed.), Nürnberg: Wiessner, (1822).
[20] Hahn, L. S., Complex numbers and geometry, Cambridge University Press, (1994).
[21] MacKay, J. S. History of the Nine Point Circle. Proceedings of the Edinburgh Mathematical Society, (1892)(11) 19-61. http://jwilson.coe.uga.edu/emt668/emt668.folders.f97/anderson/geometry/geometry1project/historyofninepointcircle/history.html
[22] Aleksa Srdanov. Is there infinity. Religija i tolerancija, Novi Sad br.26, decembar 2016.
[23] Clark Kimberling. Encyclopedia of triangles centers. [24] Aleksa Srdanov, Dragan Stojiljkovic, Counting Natural and Integer Solutions to Equations and Inequalities, International Journal of Mathematics Trends and Technology 67(7)(2021) 21-25.

Keywords : Euler line, Feuerbach circle, Monge point, Nine-point circle.