  Given a set $P$ of $n$ points in the plane, in general position, denote by
$N_\Delta(P)$ the number of empty triangles with vertices in $P$. In this paper
we investigate by how much $N_\Delta(P)$ changes if a point $x$ is removed from
$P$. By constructing a graph $G_P(x)$ based on the arrangement of the empty
triangles incident on $x$, we transform this geometric problem to the problem
of counting triangles in the graph $G_P(x)$. We study properties of the graph
$G_P(x)$ and, in particular, show that it is kite-free. This relates the growth
rate of the number of empty triangles to the famous Ruzsa-Szemer\'edi problem.
