  Recently Guth and Katz \cite{GK2} invented, as a step in their nearly
complete solution of Erd\H{o}s's distinct distances problem, a new method for
partitioning finite point sets in $\R^d$, based on the Stone--Tukey polynomial
ham-sandwich theorem. We apply this method to obtain new and simple proofs of
two well known results: the Szemer\'edi--Trotter theorem on incidences of
points and lines, and the existence of spanning trees with low crossing
numbers. Since we consider these proofs particularly suitable for teaching, we
aim at self-contained, expository treatment. We also mention some
generalizations and extensions, such as the Pach--Sharir bound on the number of
incidences with algebraic curves of bounded degree.
