Equidecomposition of Two Rectangles
What is it about?
A Mathematical Droodle

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Explanation

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A rectangle is equidecomposable with a square of equal area. It follows that two rectangles of equal area are equidecomposable. The applet illustrates a little different and probably more economical way to prove this fact. Each of the two rectangles can be split into two equal triangles and a parallelogram. In fact, all four triangles are equal, whereas the parallelograms have a common base and the height. These, as we know, are equidecomposable.

Assume the triangles have sides a, b and c, d, so that ab = cd. Assuming a < b and c < d, we also get a < d and c < b. This allows one to cut right triangles with legs a and c off each of the rectangles. The remaining parallelograms have a base √(a² + c²) and equal areas and, therefore, are equidecomposable.

References

1. T. Andreescu, B. Enescu, Mathematical Olympiad Treasures, Birkhäuser, 2004

Equidecomposition by Dissiection

• Carpet With a Hole
• Equidecomposition of a Rectangle and a Square
• Equidecomposition of Two Parallelograms
• Equidecomposition of Two Rectangles
• Equidecomposition of a Triangle and a Rectangle
• Equidecomposition of a Triangle and a Rectangle II
• Two Symmetric Triangles Are Directly Equidecomposable
  • Two Symmetric Triangles Are Directly Equidecomposable II
  • Two Symmetric Triangles Are Directly Equidecomposable III
  • Two Symmetric Triangles Are Directly Equidecomposable IV
• Wallace-Bolyai-Gerwien Theorem
• Perigal's Proof of the Pythagorean Theorem
• A Proof Perigal and All Others After Him Missed
• Dissection of a Vase
• Curvy Dissection

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