  The purpose of this article is to further explore how the structure of the
affine group can be used to deduce new incidence theorems, and to explore
sum-product type applications of these incidence bounds, building on the recent
work of Rudnev and Shkredov.
  We bound the energy of several systems of lines, in some cases obtaining a
better energy bound than the corresponding bounds obtained by Rudnev and
Shkredov by exploiting a connection with collinear quadruples.
  Our motivation for seeking to generalise and improve the incidence bound
obtained by Rudnev and Shkredov comes from possible applications to sum-product
problems. For example, we prove that, for any finite $A \subset \mathbb R$ the
following superquadratic bound holds:
  \[
  \left| \left \{ \frac{ab-cd}{a-c} : a,b,c,d \in A \right \} \right| \gg
|A|^{2+\frac{1}{14}}.
  \]
  This improves the previously known bound with exponent $2$. We also give a
threshold-beating asymmetric sum-product estimate for sets with small sum set
by proving that there exists a positive constant $c$ such that for all finite
$A,B \subset \mathbb R$,
  \[
  |A+A| \ll K|A| \Rightarrow |AB| \gg_K |A||B|^{1/2+c}.
  \]
