Squaring a Rectangle: What Is This About?
A Mathematical Droodle

Explanation

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Copyright © 1996-2012 Alexander Bogomolny

The applet suggests a construction [Heath] of a square equal in area to a given rectangle (Euclid II.14). The two shapes are equidecomposable, as are any two polygons of equal area (see Wallace-Bolyai-Gerwien Theorem.) Thus we already know how to solve the problem of squaring a rectangle. The applet shows a different, yet no less constructible, way of achieving the same goal. The construction is based on an appearance of the geometric mean in the right triangle. The applet follows Euclid VI.13 which naturally differs from the construction in Euclid II.14.

Let ABDE be a rectangle. Construct C on the extension of AB such that BC = BD. Form a semicircle on AC as a diameter. Let F be the point of intersection of BD and the semicircle. Then

which is what is required.

References

1. T. L. Heath, Euclid: The Thirteen Books of The Elements, Dover, 1956

Quadrature: A Child's Play

• Hippocrates' Squaring of a Lune
• Hippocrates' Squaring of Lunes
• Squaring a Rectangle
• Shearing a Polygon into a Triangle of Equal Area
• Triangle of Equal Area

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Copyright © 1996-2012 Alexander Bogomolny