On an application of Guth-Katz theorem

We prove that for some universal $c$, a non-collinear set of $N>\frac{1}{c}$ points in the Euclidean plane determines at least $c\frac{N}{\log N}$ distinct areas of triangles with one vertex at the origin, as well as at least $c \frac{N}{\log N}$ distinct dot products. This in particular implies a sum-product bound $$\mathrm{\mid }A\cdot A\pm A\cdot A\mathrm{\mid }\ge c\frac{\mathrm{\mid }A{\mathrm{\mid }}^2}{\log \mathrm{\mid }A\mathrm{\mid }}$$ for a discrete $A\subset \mathbb{R}$.