**Theorem 2.4**: If *C* is on the line segment between *A* and *B* then *A* and *B* are on opposite
sides of *C*.

**Proof**: Suppose that *C* is on the line segment between *A* and * B*.
Then there exists a real number* t* with

such that

If we transpose the *A* term to the left and the *C* term to the
right, we get

Since * C* is between *A* and *B*, *t* is not equal to 1, so we can divide
both sides by 1- *t* to get

Let

Moreover,

so

*u* < 0,
so *A* is on the opposite side of *C* from * B*.

If we solve *C* = (1 - *t*)*A* + *tB* for *B*, we get

Let

Then

So

*B* = (1-*u*)*A* + *uC*

Since 0 < *t* < 1, *u* > 1, and *B *is on the other side of *C *from *A*.