%%% exemples d'utilisation de la condition de colinéarité
%%% <date>2001-08-04 </date> 
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The parallelism criterion can be used to determine when a given point is on a given line. 
Let \repere be a coordinate system. 

\begin{exemple}

\item
  Let $C$ be the point the coordinates of which are \couple{1}{3}. Let \Li be the line passing through $A$:
  \couple{-1}{2} and $B$: \couple{5}{-1}. Is $C$ on $\EuScript{L}$? In fact, $C$ is on \Li if and only if \vecteur{AB} and
  \vecteur{AC} are parallel. We can determine the coordinates of \vecteur{AB} and \vecteur{AC} and then use the
  parallelism criterion (cf. \ref{vect-chap}--\ref{paralcrit}) to answer the question. 

  \vecteur{AB}: \couple{5-(-1)}{-1-2} so \vecteur{AB}: \couple{6}{-3} 
  and \vecteur{AC}: \couple{1-(-1)}{3-2} so \vecteur{AC}: \couple{2}{1}. Now as $6 \times 1 - (-3)\times 2 = 12$, we can
  state that $C$ is \textbf{not} on \Li.

\item
  Let $A$: \couple{5}{-15}, $B$: \couple{3}{-8}, and $C$: \couple{1}{-1}. The question is once more: is $C$ on $(AB)$?

  We have \vecteur{AB}: \couple{-2}{7} and \vecteur{AC}: \couple{-4}{14}. As $-2 \times 14 - 7 \times (-4) = -28 + 28 =
  0$ we can state that $C$ \textbf{is} on $(AB)$.

\item 
  Let $A$: \couple{5}{-15}, $B$: \couple{3}{-8} once again. Let $C$ be a point of $(AB)$ the abscissa of which is
  $12$. What is the ordinate of $C$?

  Let $y$ be the (unknown) ordinate of $C$. \vecteur{AB}: \couple{-2}{7} and \vecteur{AC}: \couple{7}{y+15}. As $C$ is
  on $(AB)$, $-2 \times (y+15) -7 \times 7 =0$ that is $-2y -79 =0$ and then $y=-39.5$.
\end{exemple}
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