Let $S$ be a linear ordered geometry. Fix a point $p$ and consider the pencil $\Pi(p)$ of all rays emanating from it. Let $\alpha\neq\beta \in\Pi(p)$ . A point $q$ is said to be an interior point of $\alpha$ and $\beta$ if there are points $s\in\alpha$ and $t\in\beta$ such that

1. $q$ and $s$ are on the same side of line $\line{pt}$ , and
2. $q$ and $t$ are on the same side of line $\line{ps}$ .

1. Suppose $\alpha,\beta,\rho\in\Pi(p)$ and $\rho$ is between $\alpha$ and $\beta$ . Then
   1. any point on $\rho$ is an interior point of $\alpha$ and $\beta$ ;
   2. any point on the line containing $\rho$ that is an interior point of $\alpha$ and $\beta$ must be a point on $\rho$ ;
   3. there is a point $q$ on $\rho$ that is between $\alpha$ and $\beta$ . This is known as the Crossbar Theorem, the two ``crossbars'' being $\rho$ and a line segment joining a point on $\alpha$ and a point on $\beta$ ;
   4. if $q$ is defined as above, then any point between $p$ and $q$ is between $\alpha$ and $\beta$ .
2. There are no rays between two opposite rays.
3. If $\rho$ is between $\alpha$ and $\beta$ , then $\beta$ is not between $\alpha$ and $\rho$ .
4. If $\alpha,\beta\in\Pi(p)$ has a ray $\rho$ between them, then $\alpha$ and $\beta$ must lie on the same half plane of some line.
5. The converse of the above statement is true too: if $\alpha,\beta\in\Pi(p)$ are distinct rays that are not opposite of one another, then there exist a ray $\rho\in\Pi(p)$ such that $\rho$ is between $\alpha$ and $\beta$ .
6. Given $\alpha,\beta\in\Pi(p)$ with $\alpha\neq\beta$ and $\alpha\neq-\beta$ . We can write $\Pi(p)$ as a disjoint union of two subsets:
   1. $A =\lbrace \rho\in\Pi(p)\mid \rho\mbox{ is between }\alpha\mbox{ and }\beta\rbrace$ ,
   2. $B=\Pi(p)-A$ .
   Let $\rho,\sigma\in\Pi(p)$ be two rays distinct from both $\alpha$ and $\beta$ . Suppose $x\in\rho$ and $y\in\sigma$ . Then $\rho,\sigma$ belong to the same subset if and only if $\overline{xy}$ does not intersect either $\alpha$ or $\beta$ .

Bibliography

1

D. Hilbert, Foundations of Geometry, Open Court Publishing Co. (1971)

2

K. Borsuk and W. Szmielew, Foundations of Geometry, North-Holland Publishing Co. Amsterdam (1960)

3

M. J. Greenberg, Euclidean and Non-Euclidean Geometries, Development and History, W. H. Freeman and Company, San Francisco (1974)