**Theorem
2.12**: Let * * *A*, * * *B* * * and * * *C* * * be three noncolinear points, and let

where

be a point in the
plane. Then * D* * * is on the same side of * * *BC* * * as* A* * * if and
only if * * *q* * * is positive.

**Proof**: Let

and

Then

Finally let

be the equation for the
line determined by * BC*. * * If
we check the coordinates for the point * * *D* * * in the left side
of this equation we get

Remove parentheses

These terms can be rearranged as

Since * **B* * *and * * *C* * * are on the line,

and

If we substitute these into the last equation, we get

or

However, since * q* + *r* + *s* = 1,

Our equation then becomes

or

This will satisfy the same inequality as the
point * * *A* * * if and only if * * *q* * * is positive.