Journal of Formalized Mathematics
Volume 14, 2002
University of Bialystok
Copyright (c) 2002 Association of Mizar Users

## On the General Position of Special Polygons

Mariusz Giero
University of Bialystok

### Summary.

In this paper we introduce the notion of general position. We also show some auxiliary theorems for proving Jordan curve theorem. The following main theorems are proved: \begin{enumerate} \item End points of a polygon are in the same component of a complement of another polygon if number of common points of these polygons is even; \item Two points of polygon $L$ are in the same component of a complement of polygon $M$ if two points of polygon $M$ are in the same component of polygon $L.$ \end{enumerate}

This work has been partially supported by CALCULEMUS grant HPRN-CT-2000-00102.

#### MML Identifier: JORDAN12

The terminology and notation used in this paper have been introduced in the following articles [22] [25] [19] [2] [17] [21] [15] [26] [1] [7] [5] [3] [24] [10] [4] [20] [18] [8] [9] [13] [14] [11] [12] [16] [23] [6]

#### Contents (PDF format)

1. Preliminaries
2. The Notion of General Position and Its Properties
3. Properties of Being in the Same Component of a Complement of a Polygon
4. Cells Are Convex
5. Properties of Points Lying on the Same Line
6. The Position of the Points of a Polygon with Respect to Another Polygon

#### Acknowledgments

I would like to thank Prof. Andrzej Trybulec for his help in preparation of this article. I also thank Adam Grabowski, Robert Milewski and Adam Naumowicz for their helpful comments.

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Received May 27, 2002