proof of parallelogram lawProofOfParallelogramLaw2002-05-30 21:43:572006-10-15 22:52:46Proofparallelogram law0\usepackage{graphicx}
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\newcommand{\figuraex}[2]{\begin{center}\includegraphics[#2]{#1}\end{center}}The proof follows directly from Apollonius theorem noticing that each diagonal
is a median for the triangles in which parallelogram is split by the other diagonal.
Also, the diagonals bisect each other.
\figura{parallelogramlaw}
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.