## You are here

Homebetweenness in rays

## Primary tabs

# betweenness in rays

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 $\overleftrightarrow{pt}$, 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
$\rho\in\Pi(p)$ 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 $\alpha,\beta,\rho\in\Pi(p)$ 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 $\beta$.

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. 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:

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

(b) $B=\Pi(p)-A$.

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 $\beta$.

# References

- 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)

## Mathematics Subject Classification

51F20*no label found*51G05

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections