Distinct Triangle Areas in a Planar Point Set

  • Adrian Dumitrescu
  • Csaba D. Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4513)


Erdős, Purdy, and Straus conjectured that the number of distinct (nonzero) areas of the triangles determined by n noncollinear points in the plane is at least \(\lfloor \frac{n-1}{2} \rfloor\), which is attained for ⌈n / 2⌉ and respectively \(\lfloor n/2\rfloor\) equally spaced points lying on two parallel lines. We show that this number is at least \(\frac{17}{38}n -O(1) \approx 0.4473n\). The best previous bound, \((\sqrt{2}-1)n-O(1)\approx 0.4142n\), which dates back to 1982, follows from the combination of a result of Burton and Purdy [5] and Ungar’s theorem [23] on the number of distinct directions determined by n noncollinear points in the plane.


Convex Polygon Distinct Area Point Pair Planar Point Combinatorial Geometry 
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Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • Csaba D. Tóth
    • 2
  1. 1.Deptartment of Computer Science, University of Wisconsin-Milwaukee, WI 53201-0784USA
  2. 2.Department of Mathematics, MIT, Cambridge, MA 02139USA

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