# Parallelogram law

In

mathematics , the simplest form of the**parallelogram law**belongs to elementarygeometry . It states that the sum of the squares of the lengths of the four sides of aparallelogram equals the sum of the squares of the lengths of the two diagonals. With the notation in the diagram on the right, this can be stated as: $AB^2+BC^2+CD^2+AD^2=AC^2+BD^2,.$

In case the parallelogram is a

rectangle , the two diagonals are of equal lengths and the statement reduces to thePythagorean theorem . But in general, the square of the length of "neither" diagonal is the sum of the squares of the lengths of two sides.**The parallelogram law in inner product spaces**In

inner product space s, the statement of the parallelogram law reduces to the algebraic identity:$2|x|^2+2|y|^2=|x+y|^2+|x-y|^2$

where

:$|x|^2=langle\; x,\; x\; angle.$

**Normed vector spaces satisfying the parallelogram law**Most real and complex

normed vector space s do not have inner products, but all normed vector spaces have norms (hence the name), and thus one can evaluate the expressions on both sides of "=" in the identity above. A remarkable fact is that if the identity above holds, then the norm must arise in the usual way from some inner product. Additionally, the inner product generating the norm is unique, as a consequence of thepolarization identity ; in the real case, it is given by:$langle\; x,\; y\; angle=\{|x+y|^2-|x-y|^2over\; 4\},$

or, equivalently, by

:$\{|x+y|^2-|x|^2-|y|^2over\; 2\}\; ext\{\; or\; \}\{|x|^2+|y|^2-|x-y|^2over\; 2\}.$

In the complex case it is given by

:$langle\; x,\; y\; angle=\{|x+y|^2-|x-y|^2over\; 4\}+i\{|ix-y|^2-|ix+y|^2over\; 4\}.$

**External links*** [

*http://www.unlvkappasigma.com/parallelogram_law/ The Parallelogram Law Proven Simply*] at [*http://www.unlvkappasigma.com/ UNLV Kappa Sigma*]

* [*http://www.cut-the-knot.org/Curriculum/Geometry/ParallelogramIdentity.shtml The Parallelogram Law: A Proof Without Words*] atcut-the-knot

* [*http://planetmath.org/?op=getobj&from=objects&name=ProofOfParallelogramLaw2 Proof of Parallelogram Law*] at [*http://planetmath.org/ Planet Math*]

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