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

Explanation

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 Dissection

1. Carpet With a Hole
2. Equidecomposition of a Rectangle and a Square
3. Equidecomposition of Two Parallelograms
4. Equidecomposition of Two Rectangles
5. Equidecomposition of a Triangle and a Rectangle
6. Equidecomposition of a Triangle and a Rectangle II
7. Perigal's Proof of the Pythagorean Theorem
8. Two Symmetric Triangles Are Directly Equidecomposable
9. Wallace-Bolyai-Gerwien Theorem