## Mathematics, Department of

## Date of this Version

1980

## Citation

*The American Mathematical Monthly*, Vol. 87, No. 4 (Apr., 1980), pp. 284-285

## Abstract

In [**1**] Guggenheimer states that a *Jordan polygon has two principal vertices that are exposed points of its convex hull*, and he refers to Meisters's paper [**3**]. Such a statement cannot be found in Meisters's paper, and in fact it is false. The polygon illustrated in the figure below provides a counterexample.

By a Jordan polygon P = *V*_{1} . . . *V*_{N} is meant a simple closed polygonal plane curve with *N* *sides* *V*_{l} *V*_{2}, *V*_{2} *V*_{3}, . . . , *V*_{N-1,} *V*_{N}, *V*_{N} *V*_{1}, joining the *N* *vertices* *V*_{1}, . . . ,*V*_{N}. In [**3**] any consecutive vertices *V*_{i-1}, *V*_{i}, and *V*_{i+1} of a Jordan polygon P are said to form an *ear* (regarded as the region enclosed by the triangle *V*_{i-1}* V*_{i }*V*_{i-1}) at the vertex *V*_{i} if the open chord joining *V*_{i-1 }and*V*_{i+1} lies entirely inside the polygon *P*. Two such ears are called *nonoverlapping* if the interiors of their triangular regions are disjoint. The following Two-Ears Theorem was proved in [**3**].

TWO-EARS THEOREM. *Except for triangles, every Jordan polygon has at least two* *nonoverlapping* *ears.*

## Comments

Copyright 1980 Mathematical Association of America