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
Famous quotes containing the word law:
“You are, or you are not the President of The National University Law School. If you are its President I wish to say to you that I have been passed through the curriculum of study of that school, and am entitled to, and demand my Diploma. If you are not its President then I ask you to take your name from its papers, and not hold out to the world to be what you are not.”
—Belva Lockwood (1830–1917)