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

   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 $\alpha ,\beta \in \mathrm{\Pi }(p)$ 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:

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

   2. $B=\mathrm{\Pi }(p)-A$.

   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 $\beta$.

Title	betweenness in rays
Canonical name	BetweennessInRays
Date of creation	2013-03-22 15:33:05
Last modified on	2013-03-22 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