Triangles in Euclidean Arrangements

The number of triangles in arrangements of lines and pseudolines has been object of some research Most results however concern arrangements in the projective plane In this article we add results for the number of triangles in Euclidean arrange ments of pseudolines Though the change in the embedding space from projective to Euclidean may seem small there are interesting changes both in the results and in the techniques required for the proofs In Levi proved that a nontrivial arrangement simple or not of n pseudolines in the projective plane contains n triangles To show the corresponding result for the Euclidean plane namely that a simple arrangement of n pseudolines contains n triangles we had to nd a completely di erent proof On the other hand a non simple arrangements of n pseudolines in the Euclidean plane can have as few as n triangles and this bound is best possible We also discuss the maximal possible number of triangles and some extensions Mathematics Subject Classi cations A C