## Faculty Publications, Department of Mathematics

#### Title

POLYGONS HAVE EARS

6-1975

#### Citation

The American Mathematical Monthly, Vol. 82, No. 6 (Jun. - Jul., 1975), pp. 648-651 Published by: Mathematical Association of America

#### Comments

Copyright 1975 Mathematical Association of America

#### Abstract

We refer to a simple closed polygonal plane curve with a finite number of sides as a Jordan polygon. We assume the truth of the famous Jordan Curve Theorem only for Jordan polygons. (For elementary proofs see Appendix 2 of Chapter V of [4] or Appendix B1 of [5].) Three consecutive vertices V1, V2, V3 of a Jordan polygon P = V1V2V3V4... VnV1 (n > 4) are said to form an ear (regarded as the region enclosed by the triangle V1V2V3) at the vertex V2 if the (open) chord joining V1 and V3 lies entirely inside the polygon P. We say that two ears are non-overlapping if their interior regions are disjoint; otherwise they are overlapping. If we remove or cut off an ear V1V2V3 (by drawing the chord V1V3) from the Jordan polygon P, then there remains the Jordan polygon P' = V1V3V4 ... VnV1 which has one less vertex than P.

The property of Jordan polygons expressed by the following theorem seems to provide a particularly simple and conceptual bridge from the Jordan Curve Theorem for Polygons to the Triangulation Theorem for Jordan Polygons; at least simpler perhaps than that given in Appendix B2 of [5].

COinS