Upper bounds on pairs of dot products

Given a large finite point set, $P\subset {\mathbb{R}}^2$, we obtain upper bounds on the number of triples of points that determine a given pair of dot products. That is, for any pair of positive real numbers, $(\alpha ,\beta )$, we bound the size of the set $$\{(p,q,r)\in P\times P\times P:p\cdot q=\alpha ,p\cdot r=\beta \}\mathrm{.}$$