Meisters' Two Ears Theorem

Every simple polygon with more than three vertices has at least two ears.

A polygon is a closed piecewise linear curve. A non-self-intersecting polygon is called simple. A (polygonal) ear is a triple of successive vertices A, B, C of a polygon such that AC is a diagonal that lies entirely in the interior of the polygon. B is naturally called the tip of the ear. The statement is known as Meisters' Two Ears Theorem.

Proof

References

  1. G. H. Meisters, Principal Vertices, Exposed Points, and Ears, Amer. Math. Monthly 87, 284-285, 1980

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Every simple polygon with more than three vertices has at least two ears.

Proof

The proof is based on the existence of a (diagonal) triangulation of polygons: every polygon can be split into triangles by some of its diagonals. We first establish a preliminary result:

Every triangulation of an n-gon has (n-2)-triangles formed by (n-3) diagonals.

The proof is by induction. If n = 3, the assertion is trivially true. Assume the statement holds for all n < K. Given a K-gon, find - as in the proof of the existence of a triangulation - a diagonal that splits the polygon into smaller two, say n-gon and m-gon such that n + m = K + 2 and both are less than K. Then, by the induction hypothesis, the n-gon consists of (n-2) triangles, while the m-gon consists of (m-2) triangles. In all, there are

(n - 2) + (m - 2) = (K + 2) - 4 = K - 2

triangles, as required. The number of the diagonals is

(n - 3) + (m - 3) + 1 = K + 2 - 5 = K - 3.

Now, for the proof of the main statement. Consider a triangulation of an n-gon, with n > 3. The triangulation consists of n-2 triangles. Each of the triangles in the triangulation shares at most 2 edges with the polygon. Since the latter has n edges but there are only two triangles, by the Pigeonhole Principle, there are at least two triangles with two polygon's edges. These are the ears.

(There is another proof of the theorem based on Graph Theory.)

Reference

  1. S. L. Devadoss, J. O'Rourke, Discrete and Computational Geometry, Princeton University Press, 2012

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