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


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What if applet does not run?

Explanation

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Copyright © 1996-2017 Alexander Bogomolny


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 https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

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

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Copyright © 1996-2017 Alexander Bogomolny

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