# proof of parallelogram law

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.

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.

Title proof of parallelogram law ProofOfParallelogramLaw 2013-03-22 12:41:29 2013-03-22 12:41:29 Mathprof (13753) Mathprof (13753) 5 Mathprof (13753) Proof msc 51-00 ApolloniusTheorem Median ProofOfParallelogramLaw2