proof of parallelogram law

The proof follows directly from Apollonius theorem noticing that each diagonalis a median for the triangles in which parallelogram is split by the other diagonal.Also, the diagonals bisect each other.

Therefore, Apollonius theorem implies

$$2{\left(\frac{d_1}{2}\right)}^2+{\left(\frac{d_2}{2}\right)}^2=u^2+v^2.$$

Multiplying both sides by $2$ and simplification leads to the desired expression.