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,
In case 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 joining the midpoints of the diagonals. It can be seen from the diagram that, for a parallelogram, then x = 0 and the general formula reduces to the parallelogram law.
Read more about Parallelogram Law: The Parallelogram Law in Inner Product Spaces, Normed Vector Spaces Satisfying The Parallelogram Law
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“Our law very often reminds one of those outskirts of cities where you cannot for a long time tell how the streets come to wind about in so capricious and serpent-like a manner. At last it strikes you that they grew up, house by house, on the devious tracks of the old green lanes; and if you follow on to the existing fields, you may often find the change half complete.”
—Walter Bagehot (1826–1877)