  A geometric graph is a graph drawn in the plane so that its vertices and
edges are represented by points in general position and straight line segments,
respectively. A vertex of a geometric graph is called pointed if it lies
outside of the convex hull of its neighbours. We show that for a geometric
graph with $n$ vertices and $e$ edges there are at least
$\frac{n}{2}\binom{2e/n}{3}$ pairs of disjoint edges provided that $2e\geq n$
and all the vertices of the graph are pointed. Besides, we prove that if any
edge of a geometric graph with $n$ vertices is disjoint from at most $ m $
edges, then the number of edges of this graph does not exceed
$n(\sqrt{1+8m}+3)/4$ provided that $n$ is sufficiently large.
  These two results are tight for an infinite family of graphs.
