Discrete & Computational Geometry

, Volume 3, Issue 4, pp 295–305 | Cite as

A sylvester theorem for conic sections

  • James A. Wiseman
  • Paul R. Wilson
Article

Abstract

If S is a finite set of points in the plane and no conic contains all points of S, then S determines a conic which contains exactly five points ofS.

Keywords

Discrete Comput Geom Algebraic Curve Unique Conic Conic Section Collinear Point 

References

  1. 1.
    W. E. Bonnice and L. M. Kelly, On the number of ordinary planes,J. Combin. Theory Ser. A 11 (1971), 45–53.MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    P. B. Borwein, On Sylvester's problem and Haar spaces,Pacific J. Math. 109 (1983), 275–278.MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    H. M. S. Coxeter,The Real Projective Plane, 2nd edn., Cambridge University Press, Cambridge, 1955.MATHGoogle Scholar
  4. 4.
    H. M. S. Coxeter,Introduction to Geometry, Wiley, New York, 1961. Footnote, p. 65.MATHGoogle Scholar
  5. 5.
    P. Erdös, Problem #4065,Amer. Math. Monthly 50 (1943), 65.MathSciNetCrossRefGoogle Scholar
  6. 6.
    S. Hansen, A generalization of a theorem of Sylvester on the lines determined by a finite point set,Math. Scand. 16 (1965), 175–180.MATHMathSciNetGoogle Scholar
  7. 7.
    S. Hansen,Contributions to the Sylvester-Gallai Theory, Polyteknisk Forlag, Copenhagen, 1981.Google Scholar
  8. 8.
    R. Hartshorne,Algebraic Geometry, Springer-Verlag, New York, 1977.MATHCrossRefGoogle Scholar
  9. 9.
    L. M. Kelly, Classroom notes,Amer. Math. Monthly 55 (1948), 28.MathSciNetCrossRefGoogle Scholar
  10. 10.
    L. M. Kelly and W. O. J. Moser, On the number of ordinary lines determined byn points,Canad. J. Math. 10 (1958), 210–219.MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    T. Motzkin, The lines and planes connecting the points of a finite set,Trans. Amer. Math. Soc. 70 (1951), 451–464.MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    R. Steinberg, Solution to problem #4065,Amer. Math. Monthly 51 (1944), 169–171.MathSciNetCrossRefGoogle Scholar
  13. 13.
    J. J. Sylvester, Mathematical questions and solutions, 11851,The Educatonal Times (1983), 98.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • James A. Wiseman
    • 1
  • Paul R. Wilson
    • 1
  1. 1.Department of MathematicsRochester Institute of TechnologyRochesterUSA

Personalised recommendations