{\bf \large Points and lines in a plane}


Points and lines are aspects of mathematics (or specifically of
geometry) which correspond to everyday life. This is, however, why it
is so difficult to decide where to start when discussing the field: We
all \emph{know} what lines and points are and therefore all attempts
at formal definition appear clumsy and not needed at first sight.

The situation is not quite as simple as one might think. In order to
conduct systematic investigations into geometry specific definitions
of all concepts are needed and references to common knowledge are not
adequate.

The methodology which has been selected to approach geometry is to
define \emph{points} and \emph{lines} from some of their elementary
properties, assume that some simple facts (so-called \emph{axioms})
apply and work from there. The most important axioms on points and
lines in classical geometry are:

\begin{itemize}
\item Any two distinct points define exactly one line which passes
  through both.
\item A given line and point outside the line define exactly one line
  through the point and not intersecting the original line.
\end{itemize}

\begin{wrapfigure}{l}[0cm]{0pt}
  {
\beginpicture
  \setcoordinatesystem units <0.8cm, 0.8cm> point at 0 0
  \setplotarea x from 0 to 4, y from 0 to 2
  \setlinear
  \plot 0 0 3 1 /
  \plot 0 1 3 2 /
  \put {$\bullet$} at 1.5 0.5
  \put {$\bullet$} at 1 1.33
  \put {$\bullet$} at 2 1.66
\endpicture
}

\end{wrapfigure}
These axioms can be used to derive all of the geometry which applies
to daily life.

Points in geometry have no size and lines extend infinitely in either
direction without width. If $A$ and $B$ are points and $\ell$ is a
line through the points then the part of the line between the points
is called the \emph{line seqment} or simply the \emph{seqment} from $A$ to $B$.

Normally no distinction is made between a line on the one hand and the
set of points on the line on the other hand. Through any point $A$
there are two half-lines parallel to a given line, $\ell$, one in each
direction. The point itself is on both halflines.