# Shearing a Polygon into a Triangle of Equal Area

What Is This About?

A Mathematical Droodle

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Copyright © 1996-2017 Alexander Bogomolny

### Shearing a Polygon into a Triangle of Equal Area

The applet suggests a construction [Eves] of a triangle with the area equal to that of a given convex polygon. According to the Wallace-Bolyai-Gerwien Theorem, any two polygons of equal area are equidecomposable: it is possible to cut one into polygonal pieces that can be rearranged to form the other polygon. Thus we already know how to solve the quadrature problem for (or squaring of) a convex polygon. The applet shows a different, yet no less constructible, way of achieving the same goal.

In any polygon, a triple of successive vertices, say, a, b, c, form a *ear* if Δabc lies inside the polygon. In a convex polygon, any three successive vertices constitute an ear. Let's consider three such vertices. Two of the three vertices (a and c) are joined by a diagonal of the polygon and the ear is completed by two of its sides meeting at the third vertex (b). Note that for any location of b on the line parallel to ac, the ear has the same area. For, all such triangles have the same base (ac) and the same altitude (the fixed distance between the line and ac.) Going from one such triangle to another is induced by a *shearing transformation* with the base line fixed. (Observe in passing that every simple polygon with at least four vertices has at least two ears.)

For two positions of b, b will lie on the extensions of the sides of the polygon incident to a and c. By choosing any of the two and discarding one of the vertices (a or c) we obtain a convex polygon of the same area as the given one, but with fewer vertices. The process may continue until only three vertices remain. No further reduction in the number of vertices becomes possible.

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### References

- H. Eves,
*An Introduction to the History of Mathematics*, Brooks Cole; 6 edition (January 2, 1990)

### 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

### What Is Shear Transform?

- Shearing Butterflies in Quadrilaterals
- Area of Parallelogram Formula by Shearing
- Parallelogram and Ellipses
- Proof 37 of the Pythagorean theorem - by David King
- Shearing a Polygon into a Triangle of Equal Area
- Pythagoras' Theorem By Sheer Shearing
- Shearing and Translation in Pythagorean Pants

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Copyright © 1996-2017 Alexander Bogomolny

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