  Let $P$ be a finite point set in the plane. A \emph{$c$-ordinary triangle} in
$P$ is a subset of $P$ consisting of three non-collinear points such that each
of the three lines determined by the three points contains at most $c$ points
of $P$. Motivated by a question of Erd\H{o}s, and answering a question of de
Zeeuw, we prove that there exists a constant $c>0$ such that $P$ contains a
$c$-ordinary triangle, provided that $P$ is not contained in the union of two
lines. Furthermore, the number of $c$-ordinary triangles in $P$ is
$\Omega(|P|)$.
