  A widely investigated subject in combinatorial geometry, originated from
Erd\H{o}s, is the following. Given a point set $P$ of cardinality $n$ in the
plane, how can we describe the distribution of the determined distances? This
has been generalized in many directions. In this paper we propose the following
variants. Consider planar arrangements of $n$ lines. Determine the maximum
number of triangles of unit area, maximum area or minimum area, determined by
these lines. Determine the minimum size of a subset of these $n$ lines so that
all triples determine distinct area triangles.
  We prove that the order of magnitude for the maximum occurrence of unit areas
lies between $\Omega(n^2)$ and $O(n^{9/4})$. This result is strongly connected
to both additive combinatorial results and Szemer\'edi--Trotter type incidence
theorems. Next we show a tight bound for the maximum number of minimum area
triangles. Finally we present lower and upper bounds for the maximum area and
distinct area problems by combining algebraic, geometric and combinatorial
techniques.

Combinatorics