Parallelogram Law: A Tessellation

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 below conceals, or perhaps reveals, the parallelogram identity in one of the plane tessellation suggested by the Law of Cosines. The tessellation reveals, or perhaps conceals, an octagon which in several incarnations is decomposed in various ways.

(One of the blue parallelograms and its corners 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.

Parallelogram law via plane tessellation

What if applet does not run?

Four incarnations of the octagon are explicit elsewhere.

Related material

Plane Tessellations

  • Dancing Squares or a Hinged Plane Tessellation
  • Dancing Rectangles Model Auxetic Behavior
  • A Hinged Realization of a Plane Tessellation
  • A Semi-regular Tessellation on Hinges A
  • A Semi-regular Tessellation on Hinges B
  • A Semi-regular Tessellation on Hinges C
  • Escher's Theorem
  • Napoleon Theorem by Plane Tessellation
  • Simple Quadrilaterals Tessellate the Plane
  • Pythagorean Theorem By Plane Tessellation
  • Pythagorean Theorem a la Friedrichs
  • Pythagorean Theorem By Hexagonal Tessellation
  • Hinged Greek Cross Tessellation
  • Pythagorean Theorem: A Variant of Proof by Tessellation
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