On the number of ordinary circles

We prove that any $n$ points in ${\mathbb{R}}^2$, not all on a line or circle, determine at least $\frac{1}{4}{n}^2-O(n)$ ordinary circles (circles containing exactly three of the $n$ points). The main term of this bound is best possible for even $n$. Our proof relies on a recent result of Green and Tao on ordinary lines.