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.)

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

What if applet does not run?

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

Proofs Without Words

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Copyright © 1996-2017 Alexander Bogomolny


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