Levi's Lemma, pseudolinear drawings of $K_n$, and empty triangles

Abstract: 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$(n\log} n)$ and O$(n<sup>2)$, respectively, empty triangles. All the arguments are elementary, algorithmic, and self-contained.