| Version current |
Version 2 |
| Let $S$ be a linear ordered geometry. |
Let $S$ be a linear ordered geometry. |
| Fix a point $p$ and consider the pencil $\Pi(p)$ of all rays |
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 |
emanating from it. Let $\alpha\neq\beta \in\Pi(p)$. A point $q$ is |
| said to be an \emph{interior point} of $\alpha$ and $\beta$ if there |
said to be an \emph{interior point} of $\alpha$ and $\beta$ if there |
| are points $s\in\alpha$ and $t\in\beta$ such that |
are points $s\in\alpha$ and $t\in\beta$ such that |
| \begin{enumerate} |
\begin{enumerate} |
| \item $q$ and $s$ are on the same side of line $\line{pt}$, and |
\item $q$ and $s$ are on the same side of line $\line{pt}$, and |
| \item $q$ and $t$ are on the same side of line $\line{ps}$. |
\item $q$ and $t$ are on the same side of line $\line{ps}$. |
| \end{enumerate} |
\end{enumerate} |
| A point $q$ is said to be \emph{between} $\alpha$ and $\beta$ if |
A point $q$ is said to be \emph{between} $\alpha$ and $\beta$ if |
| there are points $s\in\alpha$ and $t\in\beta$ such that $q$ is |
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 |
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 |
interior point of these rays, but not vice versa in general. A ray |
| $\rho\in\Pi(p)$ is said to be \emph{between} rays $\alpha$ and |
$\rho\in\Pi(p)$ is said to be \emph{between} rays $\alpha$ and |
| $\beta$ if there is an interior point of $\alpha$ and $\beta$ lying |
$\beta$ if there is an interior point of $\alpha$ and $\beta$ lying |
| on $\rho$. |
on $\rho$. |
| \\\\ |
\\\\ |
| \textbf{Properties} |
\textbf{Properties} |
| \begin{enumerate} |
\begin{enumerate} |
| \item Suppose $\alpha,\beta,\rho\in\Pi(p)$ and $\rho$ is between |
\item Suppose $\alpha,\beta,\rho\in\Pi(p)$ and $\rho$ is between |
| $\alpha$ and $\beta$. Then |
$\alpha$ and $\beta$. Then |
| \begin{enumerate} |
\begin{enumerate} |
| \item any point on $\rho$ is an interior point of $\alpha$ and |
\item any point on $\rho$ is an interior point of $\alpha$ and |
| $\beta$; |
$\beta$; |
| \item any point on the line containing $\rho$ that is an interior |
\item any point on the line containing $\rho$ that is an interior |
| point of $\alpha$ and $\beta$ must be a point on $\rho$; |
point of $\alpha$ and $\beta$ must be a point on $\rho$; |
| \item there is a point $q$ on $\rho$ that is between $\alpha$ and |
\item there is a point $q$ on $\rho$ that is between $\alpha$ and |
| $\beta$. This is known as the \textbf{Crossbar Theorem}, the two ``crossbars'' being $\rho$ and a line segment joining a point on $\alpha$ and a point on $\beta$; |
$\beta$. This is known as the \textbf{Crossbar Theorem}, the two ``crossbars'' being $\rho$ and a line segment joining a point on $\alpha$ and a point on $\beta$; |
| \item if $q$ is defined as above, then any point between $p$ and |
\item if $q$ is defined as above, then any point between $p$ and |
| $q$ is between $\alpha$ and $\beta$. |
$q$ is between $\alpha$ and $\beta$. |
| \end{enumerate} |
\end{enumerate} |
| \item There are no rays between two opposite rays. |
\item There are no rays between two opposite rays. |
| \item If $\rho$ is between $\alpha$ and $\beta$, then $\beta$ is not |
\item If $\rho$ is between $\alpha$ and $\beta$, then $\beta$ is not |
| between $\alpha$ and $\rho$. |
between $\alpha$ and $\rho$. |
| \item If $\alpha,\beta\in\Pi(p)$ has a ray $\rho$ between them, then |
\item 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. |
$\alpha$ and $\beta$ must lie on the same half plane of some line. |
| \item The converse of the above statement is true too: if |
\item The converse of the above statement is true too: if |
| $\alpha,\beta\in\Pi(p)$ are distinct rays that are not opposite of |
$\alpha,\beta\in\Pi(p)$ are distinct rays that are not opposite of |
| one another, then there exist a ray $\rho\in\Pi(p)$ such that $\rho$ |
one another, then there exist a ray $\rho\in\Pi(p)$ such that $\rho$ |
| is between $\alpha$ and $\beta$. |
is between $\alpha$ and $\beta$. |
| \item Given $\alpha,\beta\in\Pi(p)$ with $\alpha\neq\beta$ and |
\item Given $\alpha,\beta\in\Pi(p)$ with $\alpha\neq\beta$ and |
| $\alpha\neq-\beta$. We can write $\Pi(p)$ as a disjoint union of |
$\alpha\neq-\beta$. We can write $\Pi(p)$ as a disjoint union of |
| two subsets: |
two subsets: |
| \begin{enumerate} |
\begin{enumerate} |
| \item $A =\lbrace \rho\in\Pi(p)\mid |
\item $A =\lbrace \rho\in\Pi(p)\mid |
| \rho\mbox{ is between }\alpha\mbox{ and }\beta\rbrace$, |
\rho\mbox{ is between }\alpha\mbox{ and }\beta\rbrace$, |
| \item $B=\Pi(p)-A$. |
\item $B=\Pi(p)-A$. |
| \end{enumerate} |
\end{enumerate} |
| Let $\rho,\sigma\in\Pi(p)$ be two rays distinct from both $\alpha$ |
Let $\rho,\sigma\in\Pi(p)$ be two rays distinct from both $\alpha$ |
| and $\beta$. Suppose $x\in\rho$ and $y\in\sigma$. Then |
and $\beta$. Suppose $x\in\rho$ and $y\in\sigma$. Then |
| $\rho,\sigma$ belong to the same subset if and only if |
$\rho,\sigma$ belong to the same subset if and only if |
| $\overline{xy}$ does not intersect either $\alpha$ or $\beta$. |
$\overline{xy}$ does not intersect either $\alpha$ or $\beta$. |
| \end{enumerate} |
\end{enumerate} |
|
|
| \begin{thebibliography}{6} |
\begin{thebibliography}{6} |
| \bibitem{dh} D. Hilbert, {\it Foundations of Geometry}, Open Court Publishing Co. (1971) |
\bibitem{dh} D. Hilbert, {\it Foundations of Geometry}, Open Court Publishing Co. (1971) |
| \bibitem{bs} K. Borsuk and W. Szmielew, {\it Foundations of Geometry}, North-Holland Publishing Co. Amsterdam (1960) |
\bibitem{bs} K. Borsuk and W. Szmielew, {\it Foundations of Geometry}, North-Holland Publishing Co. Amsterdam (1960) |
| \bibitem{mg} M. J. Greenberg, {\it Euclidean and Non-Euclidean Geometries, Development and History}, W. H. Freeman and Company, San Francisco (1974) |
\bibitem{mg} M. J. Greenberg, {\it Euclidean and Non-Euclidean Geometries, Development and History}, W. H. Freeman and Company, San Francisco (1974) |
| \end{thebibliography} |
\end{thebibliography} |
