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.
- G. H. Meisters, Principal Vertices, Exposed Points, and Ears, Amer. Math. Monthly 87, 284-285, 1980
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.)
- S. L. Devadoss, J. O'Rourke, Discrete and Computational Geometry, Princeton University Press, 2012