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

{\displaystyle 2AB^{2}+2BC{2}=AC^{2}+BD{2},}{\displaystyle 2AB^{2}+2BC{2}=AC^{2}+BD{2},}
If the parallelogram is a rectangle, the two diagonals are of equal lengths AC = BD, so

{\displaystyle 2AB^{2}+2BC{2}=2AC^{2}}{\displaystyle 2AB^{2}+2BC{2}=2AC^{2}}
and the statement reduces to the Pythagorean theorem. For the general quadrilateral with four sides not necessarily equal,
{\displaystyle AB^{2}+BC{2}+CD^{2}+DA{2}=AC^{2}+BD{2}+4x^{2},}{\displaystyle AB^{2}+BC{2}+CD^{2}+DA{2}=AC^{2}+BD{2}+4x^{2},}
where {\displaystyle x}x is the length of the line segment joining the midpoints of the diagonals. It can be seen from the diagram that {\displaystyle x=0}x=0 for a parallelogram, and so the general formula simplifies to the parallelogram law.

A parallelogram. The sides are shown in blue and the diagonals in red.

1 Proof

2 The parallelogram law in inner product spaces

3 Normed vector spaces satisfying the parallelogram law

4 See also