Triangles in Euclidean arrangements

The number of triangles in arrangements of lines and pseudolines has been t he object of some research. Most results, however, concern arrangements in the projective plane. In this article we add results for the number of tria ngles in Euclidean arrangements of pseudolines. Though the change in the em bedding space from projective to Euclidean may seem small there are interes ting changes both in the results and in the techniques required for the pro ofs. In 1926 Levi proved that a nontrivial arrangement-simple or not-of n pseudo lines in the projective plane contains at least n triangles. To show the co rresponding result for the Euclidean plane, namely, that a simple arrangeme nt of n pseudolines contains at least n - 2 triangles, we had to find a com pletely different proof. On the other hand a nonsimple arrangement of n pse udolines in the Euclidean plane can have as few as 2n/3 triangles and this bound is best possible. We also discuss the maximal possible number of tria ngles and some extensions.