An axiomatic look at the Erdős-Trost problem

Franz Kalhoff, Victor Pambuccian

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

The Erdős-Trost problem can be formulated in the following way: “If the triangle XY Z is inscribed in the triangle ABC—with X, Y, and Z on the sides BC, CA, and AB, respectively—then one of the areas of the triangles BXZ, CXY , AY Z is less than or equal to the area of the triangle XY Z.” There are many different solutions for this problem. In this note we take up a very elementary proof (due to Szekeres) and deduce that the class of ordered translation planes is the level in the hierarchy of affine planes where the Erdős-Trost statement still holds true. We also look at the conditions an absolute plane needs to satisfy for the validity of the Erdős-Trost statement.

Original languageEnglish (US)
Pages (from-to)379-385
Number of pages7
JournalJournal of Geometry
Volume107
Issue number2
DOIs
StatePublished - Jul 1 2016

Keywords

  • area inequality
  • Hilbert planes
  • Ordered translation planes

ASJC Scopus subject areas

  • Geometry and Topology

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