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

0 < t < 1

such that

C = (1 - t)A + tB

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

(1 - t)A = C - tB

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

A = (1 - u)C + uB

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.

next theorem (2.5)