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# 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.

Related:

ApolloniusTheorem,Median, ProofOfParallelogramLaw2

Type of Math Object:

Proof

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Reference

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## Mathematics Subject Classification

51-00*no label found*

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