Shearing a Polygon into a Triangle of Equal Area

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.

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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