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

C = (1 - t)A + tB

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

(t - 1)A = tB - C

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

Let

Moreover,

so

A = (1 - u)C + uB

Since t is between 0 and 1, t is positive and t - 1 is negative, so u is negative, and A is on the opposite side of C from B.

next theorem (2.5)