Checking whether a point lies on a wide line segment I know how to define a point on the segment, but I have this piece is a wide interval. The line has a width.
I have $x_1$ $y_1$, $x_2$ $y_2$, width and $x_3$ $y_3$
$x_3$ and $x_4$ what you need to check.
perhaps someone can help, and function in $\Bbb{C}$ #

Trying to understand your question, perhaps this picture might help.

You seem to be asking how to find out whether the point $C$ is inside the thick line $AB$.
You should drop a perpendicular from $C$ to $AB$, meeting at the point $D$.  If the (absolute) length of $CD$ is more than half the width of the thick line then $C$ is outside the thick line (as shown in this particular case).  
If the thick line is in fact a thick segment, then you also have to consider whether $D$ is between $A$ and $B$ (or perhaps slightly beyond one of them, if the thickness extends further).  
