# Parallelogram law

In mathematics, the simplest form of the **parallelogram law** (also called the **parallelogram identity**) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. Using the notation in the diagram on the right, the sides are (*AB*), (*BC*), (*CD*), (*DA*). But since in Euclidean geometry a parallelogram necessarily has opposite sides equal, or (*AB*) = (*CD*) and (*BC*) = (*DA*), the law can be stated as,

If the parallelogram is a rectangle, the two diagonals are of equal lengths (*AC*) = (*BD*) so,

and the statement reduces to the Pythagorean theorem. For the general quadrilateral with four sides not necessarily equal,

where *x* is the length of the line segment joining the midpoints of the diagonals. It can be seen from the diagram that, for a parallelogram, *x* = 0, and the general formula simplifies to the parallelogram law.

## Proof

In the parallelogram on the left, let AD=BC=a, AB=DC=b, ∠BAD = α. By using the law of cosines in triangle ΔBAD, we get:

In a parallelogram, adjacent angles are supplementary, therefore ∠ADC = 180°-α. By using the law of cosines in triangle ΔADC, we get:

By applying the trigonometric identity to the former result, we get:

Now the sum of squares can be expressed as:

After simplifying this expression, we get:

## The parallelogram law in inner product spaces

In a normed space, the statement of the parallelogram law is an equation relating norms:

In an inner product space, the norm is determined using the inner product:

As a consequence of this definition, in an inner product space the parallelogram law is an algebraic identity, readily established using the properties of the inner product:

Adding these two expressions:

as required.

If *x* is orthogonal to *y*, then and the above equation for the norm of a sum becomes:

which is Pythagoras' theorem.

## Normed vector spaces satisfying the parallelogram law

Most real and complex normed vector spaces do not have inner products, but all normed vector spaces have norms (by definition). For example, a commonly used norm is the *p*-norm:

where the are the components of vector .

Given a norm, one can evaluate both sides of the parallelogram law above. A remarkable fact is that if the parallelogram law holds, then the norm must arise in the usual way from some inner product. In particular, it holds for the *p*-norm if and only if *p* = 2, the so-called *Euclidean* norm or *standard* norm.[1][2]

For any norm satisfying the parallelogram law (which necessarily is an inner product norm), the inner product generating the norm is unique as a consequence of the polarization identity. In the real case, the polarization identity is given by:

or, equivalently, by:

In the complex case it is given by:

For example, using the *p*-norm with *p* = 2 and real vectors , the evaluation of the inner product proceeds as follows:

which is the standard dot product of two vectors.

## References

- Cantrell, Cyrus D. (2000).
*Modern mathematical methods for physicists and engineers*. Cambridge University Press. p. 535. ISBN 0-521-59827-3.if

*p*≠ 2, there is no inner product such that because the*p*-norm violates the parallelogram law. - Saxe, Karen (2002).
*Beginning functional analysis*. Springer. p. 10. ISBN 0-387-95224-1.

## External links

- Weisstein, Eric W. "Parallelogram Law".
*MathWorld*. - The Parallelogram Law Proven Simply at Dreamshire blog
- The Parallelogram Law: A Proof Without Words at cut-the-knot
- A generalization of the "Parallelogram Law/Identity" to a Parallelo-hexagon and to 2n-gons in General - Relations between the sides and diagonals of 2n-gons (Douglas' Theorem) at Dynamic Geometry Sketches, an interactive dynamic geometry sketch.