Theorem 2.12: Let A, B and C be three noncolinear points, and let
be a point in the plane. Then D is on the same side of BC as A if and only if q is positive.
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
These terms can be rearranged as
Since B and C are on the line,
If we substitute these into the last equation, we get
However, since q + r + s = 1,
Our equation then becomes
This will satisfy the same inequality as the point A if and only if q is positive.
next theorem (2.13)