**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

**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

and

There are three cases: