Definitions

1. If $a$ and $b$ are distinct points, the line $ab$ is the set of all points $p$ such that $p*a*b$ or $a*p*b$ or $a*b*p$ . It can be shown that the line $ab$ and the line $ba$ are the same set of points.
2. If $o$ and $a$ are distinct points, a ray $[oa$ is the set of all points $p$ such that $p=o$ or $o*p*a$ or $o*a*p$ .
3. If $a$ and $b$ are distinct points, the open interval is the set of points $p$ such that $a*p*b$ . It is denoted by $(a,b).$
4. If $a$ and $b$ are distinct points, the closed interval is $(a,b) \cup \{a\} \cup \{b\}$ , and denoted by $[a,b].$
5. The way $a_1a_2\ldots a_n$ is the finite set of points $\{a_1, \ldots , a_n\}$ along with the open intervals $(a_1, a_2), (a_2,a_3), \ldots, (a_{n-1}, a_n)$ . The points $a_1, \ldots, a_n$ are called the vertices of the way, and the open intervals are called the sides of the way. A way is also called a broken line. The closed intervals $[a_1,a_2], \ldots, [a_{n-1},a_n]$ are called the side-intervals of the way. The lines $a_1a_2, \ldots , a_{n-1}a_n$ are called the side-lines of the way. The way $a_1a_2\ldots a_n$ is said to join $a_1$ to $a_n$ . It is assumed that $a_{i-1}, a_i, a_{i+1}$ are not collinear.
6. A way is said to be simple if it does not meet itself. To be precise, (i) no two side-intervals meet in any point which is not a vertex, and (ii) no three side-intervals meet in any point.
7. A polygon is a way $a_1 a_2 \ldots a_n$ for which $a_1 = a_n$ . Notice that there is no assumption that the points are coplanar.
8. A simple polygon is polygon for which the way is simple.
9. A region is a set of points not all collinear, any two of which can be joined by points of a way using only points of the region.
10. A region $R$ is convex if for each pair of points $a,b \in R$ the open interval $(a,b)$ is contained in $R.$
11. Let $X$ and $Y$ be two sets of points. If there is a set of points $S$ such that every way joining a point of $X$ to a point of $Y$ meets $S$ then $S$ is said to separate $X$ from $Y$ .
12. If $a_1 a_2 \ldots a_n$ is a polygon, then the angles of the polygon are $\angle a_na_1a_2, \angle a_1a_2a_3$ , and so on.

Now assume that all points of the geometry are in one plane. Let $P$ be a polygon. ($P$ is called a plane polygon.)

1. A ray or line which does not go through a vertex of $P$ will be called suitable.
2. An inside point $a$ of $P$ is one for which a suitable ray from $a$ meets $P$ an odd number of times. Points that are not on or inside $P$ are said to be outside $P$ .
3. Let $\{P_i\}$ be a set of polygons. We say that $\{P_i\}$ dissect $P$ if the following three conditions are satisfied: (i) $P_i$ and $P_j$ do not have a common inside point for $i \not = j$ , (ii) each inside point of $P$ is inside or on some $P_i$ and (iii) each inside point of $P_i$ is inside $P$ .
4. A convex polygon is one whose inside points are all on the same side of any side-line of the polygon.

Theorems

1. It can be shown that $P$ separates the other points of the plane into at least two regions and that if $P$ is simple there are exactly two regions. Moise proves this directly in [3], pp. 16-18.
2. It can be shown that $P$ can be dissected into triangles $\{T_i\}$ such that every vertex of a $T_i$ is a vertex of $P$ .
3. The following theorem of Euler can be shown: Suppose $P$ is dissected into $f>1$ polygons and that the total number of vertices of these polygons is $v$ , and the number of open intervals which are sides is $e$ . Then $$ v-e+f = 1 $$ .

A plane simple polygon with $n$ sides is called an $n$ -gon, although for small $n$ there are more traditional names:

A plane simple polygon is also called a Jordan polygon.

Bibliography

1

K. Borsuk and W. Szmielew, Foundations of Geometry, North-Holland Publishing Company, 1960.

2

H.G. Forder, The Foundations of Euclidean Geometry, Dover Publications, 1958.

3

E.E. Moise, Geometric Topology in Dimensions 2 and 3, Springer-Verlag, 1977.