## Mathematics, Department of

#### Title

#### Date of this Version

6-1975

#### Citation

*The American Mathematical Monthly*, Vol. 82, No. 6 (Jun. - Jul., 1975), pp. 648-651 Published by: 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 *V*_{1}, *V*_{2}, *V*_{3} of a Jordan polygon P = *V _{1}V_{2}V_{3}V_{4}... V*

_{n}

*V*

_{1}(n > 4) are said to form an

*ear*(regarded as the region enclosed by the triangle

*V*) at the vertex

_{1}V_{2}V_{3}*V*

_{2}if the (open) chord joining

*V*and V

_{1}_{3}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

*V*(by drawing the chord

_{1}V_{2}V_{3}*V*) from the Jordan polygon

_{1}V_{3}*P*, then there remains the Jordan polygon

*P*' =

*V*...

_{1}V_{3}V_{4}*V*

_{n}

*V*

_{1}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**].

## Comments

Copyright 1975 Mathematical Association of America