Journal of Geometry

, Volume 63, Issue 1–2, pp 39–56 | Cite as

Locus properties of the Neuberg cubic

  • Zvonko Čerin


In this paper we explore various locus problems whose solutions involve the Neuberg cubic of the scalene triangle in the plane. We use analytical geometry to show that the Neuberg equation describes the essential part of the locus in each of these problems. In this way we discover new characteristics of the Neuberg cubic that has been at the focus of attention in the recent renaissance of triangle geometry.


Locus Problem Locus Property Analytical Geometry Scalene Triangle Triangle Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bresson,Question 2119, Mathesis (1922), 431.Google Scholar
  2. [2]
    Bresson and M.Legros,Question 2119, Mathesis (1923), 178–180.Google Scholar
  3. [3]
    B. H. Brown,The 21-point cubic, Amer. Math. Monthly,32 (1925), 110–115.Google Scholar
  4. [4]
    B. H. Brown,Note on the preceding paper, Amer. Math. Monthly,32 (1925), 247.Google Scholar
  5. [5]
    Z.Čerin,Locus of intersections of Euler lines, Rad HAZU, (Preprint).Google Scholar
  6. [6]
    Z. Čerin,An extraordinary locus recognised, Math, and Inf. Quar.6 (1996), 101–106.Google Scholar
  7. [7]
    Z.Čerin,On the cubic of Napoleon, Journal of Geometry, (to appear).Google Scholar
  8. [8]
    H. S. M. Coxeter,Cubic Curves related to a Quadrangle, C. R. Math. Rep. Acad. Sci. Canada15 (1993), 237–242.Google Scholar
  9. [9]
    H. S. M. Coxeter,Some applications of trilinear coordinates, Linear Algebra and Its Applications226–228 (1995), 375–388.Google Scholar
  10. [10]
    H. M. Cundy andC. F. Parry,Some cubic curves associated with a triangle, Journal of Geometry,53 (1995), 41–66.Google Scholar
  11. [11]
    R. Deaux,Introduction to the geometry of complex numbers, Ungar Publishing Co., New York, 1956.Google Scholar
  12. [12]
    H. L. Dorwart,The Neuberg cubic: A nostalgic look, California Mathematics3 (1978), 31–38.Google Scholar
  13. [13]
    R. H. Eddy andJ. B. Wilker,Plane mappings of isogonal-isotomic type, Soochow Journal of Math.18 (1992), 123–126.Google Scholar
  14. [14]
    L. Hahn,Complex numbers and geometry, MAA, Washington, 1994.Google Scholar
  15. [15]
    E.Hain,Question 653, Mathesis (1890),22.Google Scholar
  16. [16]
    Ross Honsberger,Episodes in nineteenth and twentieth century Euclidean geometry, MAA, New Mathematical Library no. 37, Washington, 1995.Google Scholar
  17. [17]
    R. A. Johnson,Advanced Euclidean Geometry, Dover Publ., Washington, 1964.Google Scholar
  18. [18]
    C. Kimberling,Central points and central lines in the plane of a triangle, Mathematics Magazine,67 (1994), 163–187.Google Scholar
  19. [19]
    T. W. Moore andJ. H. Neelley,The circular cubic on twenty-one points of a triangle, Amer. Math. Monthly,32 (1925), 241–246.Google Scholar
  20. [20]
    F. Morley,Note on Neuberg cubic curve, Amer. Math. Monthly,32 (1925), 407–411.Google Scholar
  21. [21]
    F. Morley andF. V. Morley,Inversive Geometry, Chelsea Publ. Co., New York, 1954.Google Scholar
  22. [22]
    J. Neuberg,Memoire sur la tetraedre, F. Hayez, Bruxelles, 1884.Google Scholar
  23. [23]
    J.Neuberg,Sur la parabole de Kiepert, Ann. de la Soc. sci. de Bruxelles, (1909-1910), 1–11.Google Scholar
  24. [24]
    J. Neuberg,Bibliographie du triangle et du tétraèdre, Mathesis,38 (1924), 241.Google Scholar
  25. [25]
    D. Pedoe,A course of geometry, Cambridge Univ. Press, Cambridge, 1970.Google Scholar
  26. [26]
    G. M. Pinkernell,Cubic curves in the triangle plane, Journal of Geometry55 (1996), 141–161.Google Scholar
  27. [27]
    P. Rubio,Cubic lines relative to a triangle, Journal of Geometry34 (1989), 152–171.Google Scholar
  28. [28]
    H. Schwerdtfeger,Geometry of complex numbers, Oliver and Boyd, Toronto, 1962.Google Scholar
  29. [29]
    J. Tabov,An extraordinary locus, Math, and Inf. Quar.,4 (1994), 70.Google Scholar
  30. [30]
    O. Thalberg,Application of the theorem of residuation to the 21-point cubic, Amer. Math. Monthly,32 (1925), 412–414.Google Scholar
  31. [31]
    I. M. Yaglom,Complex numbers in geometry, Academic Press, New York, 1968.Google Scholar
  32. [32]
    P. Yff,Two families of cubics associated with a triangle, In Eves' Circles (Joby Milo Anthony, editor), MAA, Washington, 1993.Google Scholar

Copyright information

© Birkhäuser Verlag 1998

Authors and Affiliations

  • Zvonko Čerin
    • 1
  1. 1.ZagrebCroatia

Personalised recommendations