**Theorem
2.7**: Given
points * A* and *B* and a line whose equation is *ax* + *by* = *c*, where * A* is either on the line
or on the same side of the
line as *B*, every point on the
line segment between *A* and *B* is
on the same side of the
line as *B*.

**Proof**: Let * A* = (*x*_{1}, *y*_{1}) and let *B* =
(*x*_{2}, *y*_{2}).

The equation of the
line is *ax* + *by* = *c*. We can
assume that

because if we find the inequality going the other way, the proof will be similar. Since * A* is supposed to be
either on the line or on
the same side as *B*, we have

Let * C* be a point on the line
segment between *A* and * B*. Then by Theorem
2.1 there is a real number *t* such that

and by the definition of a line
segment, 0 < *t* < 1.

If we substitute these coordinates into the left side of the equation for the line, we get

But since 0 < *t* < 1, both *t* and 1 - *t* are positive so
this is

and *C* is on the same side
of the line as *B*.