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.

We use these notations for the sides: AB, BC, CD, DA.

But since in Euclidean geometry a parallelogram necessarily has opposite sides equal, that is, 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) Euler's quadrilateral theorem states

is the length of the line segment joining the midpoints of the diagonals.

In the parallelogram on the right, let AD = BC = a, AB = DC = b,

By using the law of cosines in triangle

In a parallelogram, adjacent angles are supplementary, therefore

Using the law of cosines in triangle

By applying the trigonometric identity

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

The parallelogram law is equivalent to the seemingly weaker statement:

because the reverse inequality can be obtained from it by substituting

With the same proof, the parallelogram law is also equivalent to:

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:

and the above equation for the norm of a sum becomes:

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 for a vector

in the real coordinate space

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.

[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:

the evaluation of the inner product proceeds as follows:

which is the standard dot product of two vectors.

Another necessary and sufficient condition for there to exist an inner product that induces the given norm

is for the norm to satisfy Ptolemy's inequality:[3]