Theorem 2.10: Let   A,   B,   and   C   be three noncolinear points. If   D   is on the line through   A   which is parallel to   BC,   then there is a real number   s   such that

D = A + s(C - B)

Proof: If   A = D,   then we can take   s = 0,   and we're done. So we can assume that   D   is a different point than   A.

Let

A = (x0, y0)

B = (x1, y1)

C = (x2, y2)

and

D = (x, y)

There are three cases:

next theorem (2.11)