**Theorem 2.3**: If * C* is on the line segment between *A* and *B* then

If *C* is on the line
determined by * A* and *B* but on the other side of *B* from * A* then

If *C* is on the line
determined by *A* and *B* but on the other side of *A* from *B*,
then

**Proof**: If *C* is on the line determined by *A* and *B*, then by Theorem 2.1, There is a real
number* t* such that

and by Theorem 2.2,

and

In any event,

If *C* is on the line segment
between *A* and *B,* *t* is between 0 and 1, and *t* and 1 - *t* will
both be positive and so we could drop the absolute value signs. In
that case

but if *t* larger than 1 or less than 0, then when you add the two
quantities above, one of them will be negative depending on which side of * A* or *B* *C* lies, and the results will follow.