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. 1.

$q$ and $s$ are on the same side of line $\overleftrightarrow{pt}$, and

2. 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. 1.

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

1. (a)

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

2. (b)

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

3. (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 $\beta$;

4. (d)

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

2. 2.

There are no rays between two opposite rays.

3. 3.

If $\rho$ is between $\alpha$ and $\beta$, then $\beta$ is not between $\alpha$ and $\rho$.

4. 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. 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. 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)

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

2. (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)
 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