**Theorem
2.8**: If a line segment
contains points on both
sides of another
line, then the
line must intersect the
segment somewhere between its
endpoints.

**Proof**: Let the two points be *A* = (*x*_{1}, *y*_{1}) and *B* = (*x*_{2}, *y*_{2}), and let the line be given by the equation *ax* + *by* = *c*.

If *A *is on one side and *B *is on the other, say

*ax*_{1} + *by*_{1} < c and *ax*_{2} + *by*_{2} > *c*

Let

*r* = *c* - (*ax*_{1} + *by*_{1})*ax*_{2} + *by*_{2 }and s = (*ax*_{2} + *by*_{2}) - *c*

Note, both *r* and *s* are positive.

Let

Note, 0 < *t* < 1, and

Then

*C* = (1 - *t*)*A* + *tB*

is on the line segment between *A* and *B* by Theorem 2.1

*C* = (1 - *t*)(*x*_{1}, *y*_{1}) + t(*x*_{2}, *y*_{2})

Let us substitute the coordinate of *C* into the equation of the line.

= *c*

and we see that point *C* is on the line.

If the line determined by the segment doesn't cross the line then by Theorem 1.6, they are parallel. But if they are parallel, then by Theorem 2.6, all of the points of the segment are on the same side of the line, which contradicts the hypotheses. Since the lines are not parallel, then by Theorem 1.6, there will be a point of intersection between the two lines.

Suppose the point of
intersection is not between the
endpoints of the
segment. Let *A* and *B* be the endpoints of the
segment, and let *C* be the point of intersection. We
can assume that * A* and *B* have been labeled so that *C* is on the opposite side of * A* from *B*.

Then

for some negative real number *t* by Theorem
2.1 and the definition of the
opposite side of *A* from * B*.

Then

Since *t* is negative, 1 - *t* is greater than 1. Since it is not
0, we can divide both sides by it and get

If we let

then

So *A* is of the form

and since 1 - *t* is greater than 1, *u* is between 0 and 1, so *A* is
on the segment between * B* and *C*.

In particular, *A* and * B* are on the same side of * C*, so by Theorem 2.5 they are on the
same side of the
line. By
Theorem 2.7 any point on the line segment between them will be
on the same side of the
line contradicting the
hypothesis.