An axiomatic look at the Erdős-Trost problem

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.