Parallelogram Law

The Parallelogram Law plays a fundamental role in higher mathematics. In a vector space endowed with a scalar product u.v, the norm ||·|| that is defined as ||u||2 = u.u, satisfies the Parallelogram Law:

(1) ||u - v||2 + ||u + v||2 = 2(||u||2 + ||v||2).

The importance of the law stems from the fact that any norm that satisfies (1) is bound to have been originated from a scalar product. For example, in a real vector space, the scalar product can be defined by

u.v = (||u + v||2 - ||u - v||2)/4, or
u.v = (||u + v||2 - ||u||2 - ||v||2)/2.

The real product defined for two complex numbers is just the common scalar product of two vectors. In plane geometry the interpretation of the parallelogram law is simple that the sum of squares formed on the diagonals of a parallelogram equal the sum of squares formed on its four sides. In plane geometry (1) is an easy consequence of the Law of Cosines. When parallelogram turns out to be a rectangle, its diagonals coincide so that (1) reduces to the Pythagorean theorem.

The applet below serves as a proof without words for (1) [Wise].

(In the applet, the corners of the reference parallelogram are draggable as is the parallelogram itself.)

The parallelogram law is related to a problem by V. Thébault. There is an additional demonstration of the law and even a better one.

Reference

  1. R. B. Nelsen, Proofs Without Words II, MAA, 2000, p. 9.
  2. D. S. Wise, Proof without Words: A Generalization from Pythagoras, Math Magazine, v. 71, no 1 (Feb., 1998), p. 64.
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