# Meisters' Two Ears Theorem

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*.

### References

- G. H. Meisters,
__Principal Vertices, Exposed Points, and Ears__,*Amer. Math. Monthly*87, 284-285, 1980

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Copyright © 1996-2018 Alexander Bogomolny### 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 - 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

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

### Reference

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

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Copyright © 1996-2018 Alexander Bogomolny