Levi's Lemma, pseudolinear drawings of K n, and empty triangles

There are three main thrusts to this article: a new proof of Levi's Enlargement Lemma for pseudoline arrangements in the real projective plane; a new characterization of pseudolinear drawings of the complete graph; and proofs that pseudolinear and convex drawings of K n have n 2 +O(n logn ) and O(n2), respectively, empty triangles. All the arguments are elementary, algorithmic, and self-contained.