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
Canonical name	ProofOfParallelogramLaw
Date of creation	2013-03-22 12:41:29
Last modified on	2013-03-22 12:41:29
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	5
Author	Mathprof (13753)
Entry type	Proof
Classification	msc 51-00
Related topic	ApolloniusTheorem
Related topic	Median
Related topic	ProofOfParallelogramLaw2