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.
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:
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