Condition for a point to lie between two lines

Question

Consider a point $(x,y)$ and two lines $\ell_1, \ell_2$. What is the condition in $x$ and $y$, which ensures that $(x,y)$ lies in the area between the two lines?

Answer 1

If the two lines coincide the question is meaningless; if the two lines intersect in one unique point then any point not on any of the two lines is "between them". Finally, if the two lines are different and parallel then there seems to be some definite meaning to "between the lines" : let the lines be

$$l_1: y=mx+n\;\;,\;\;y=mx+n'$$

Then a point $\,(a,b)\in\Bbb R^2\,$ is in between the lines (my definition now) if it is both in the semiplane determined by the first line and in the semiplane determined by the second one, i.e.: supposing that $\,n>n'\,$ , then

$$ma+n'< b< ma+n\Longleftrightarrow (a,b)\in l_3$$

for some line $\,l_3: y= mx+r\,\,,\,\,n'<r<n\,$ (**) .

In the above, the point is assumed not to lie in any of the line $\,l_1,l_2\,$ but actually "between them". If you want to include both lines in the above definition simply take weak inequalities in (**) above.