Parallelogram Law: A PWW

Parallelogram Law asserts that the sum of squares formed on the diagonals of a parallelogram equals the sum of squares formed on its sides. The applet is a realization of an idea first conceived by Douglas Rogers. A heptagon that appears in a proof of the Law of Cosines is naturally expanded to an octagon with a four-fold rotational symmetry. The octagon is decomposed in four ways which combine to suggest a proof (without words) of the parallelogram law.

(The blue parallelogram in the upper left corner and its vertices are draggable.)

The octagon, with its decompositions, emerges in a related plane tessellation.

Copyright © 1996-2010 Alexander Bogomolny