betweenness in rays
Let $S$ be a linear ordered geometry. Fix a point $p$ and consider the pencil $\mathrm{\Pi}(p)$ of all rays emanating from it. Let $\alpha \ne \beta \in \mathrm{\Pi}(p)$. 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 $\overleftrightarrow{pt}$, and

2.
$q$ and $t$ are on the same side of line $\overleftrightarrow{ps}$.
A point $q$ is said to be between $\alpha $ and $\beta $ if
there are points $s\in \alpha $ and $t\in \beta $ such that $q$ is
between $s$ and $t$. A point that is between two rays is an
interior point of these rays, but not vice versa in general. A ray
$\rho \in \mathrm{\Pi}(p)$ is said to be between rays $\alpha $ and
$\beta $ if there is an interior point of $\alpha $ and $\beta $ lying
on $\rho $.
Properties

1.
Suppose $\alpha ,\beta ,\rho \in \mathrm{\Pi}(p)$ and $\rho $ is between $\alpha $ and $\beta $. Then

(a)
any point on $\rho $ is an interior point of $\alpha $ and $\beta $;

(b)
any point on the line containing $\rho $ that is an interior point of $\alpha $ and $\beta $ must be a point on $\rho $;

(c)
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 $;

(d)
if $q$ is defined as above, then any point between $p$ and $q$ is between $\alpha $ and $\beta $.

(a)

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 $\alpha ,\beta \in \mathrm{\Pi}(p)$ 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 $\alpha ,\beta \in \mathrm{\Pi}(p)$ are distinct rays that are not opposite of one another, then there exist a ray $\rho \in \mathrm{\Pi}(p)$ such that $\rho $ is between $\alpha $ and $\beta $.

6.
Given $\alpha ,\beta \in \mathrm{\Pi}(p)$ with $\alpha \ne \beta $ and $\alpha \ne \beta $. We can write $\mathrm{\Pi}(p)$ as a disjoint union^{} of two subsets:

(a)
$A=\{\rho \in \mathrm{\Pi}(p)\mid \rho \text{is between}\alpha \text{and}\beta \}$,

(b)
$B=\mathrm{\Pi}(p)A$.
Let $\rho ,\sigma \in \mathrm{\Pi}(p)$ 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 $.

(a)
References
 1 D. Hilbert, Foundations of Geometry, Open Court Publishing Co. (1971)
 2 K. Borsuk and W. Szmielew, Foundations of Geometry, NorthHolland Publishing Co. Amsterdam (1960)
 3 M. J. Greenberg, Euclidean^{} and NonEuclidean Geometries, Development^{} and History, W. H. Freeman and Company, San Francisco (1974)
Title  betweenness in rays 
Canonical name  BetweennessInRays 
Date of creation  20130322 15:33:05 
Last modified on  20130322 15:33:05 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  6 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 51F20 
Classification  msc 51G05 
Related topic  Angle 
Related topic  Ray 
Related topic  Midpoint4 
Defines  interior point 
Defines  between rays 
Defines  between two rays 
Defines  crossbar theorem 