Theorem 2.7: Given a points A and B where A which is either on the line or on the same side of the line as B, every point on the line segment between A and B is on the same side of the line as B.

Proof: Let A = (x₁, y₁) and let B = (x₂, y₂).

Let the equation of the line be ax + by = c. We can assume that

because if we find the inequality going the other way, we could multiply both sides by a negative number. Since A is supposed to be either on the line or on the same side as B, we have

Let C be a point on the line segment between A and B. Then by Theorem 2.1 there is a real number t such that

and by the definition of a line segment, t is between 0 and 1.

If we substitute these coordinates into the left side of the equation for the line, we get

But since t is between 0 and 1, both t and 1 - t are positive so this is

and C is on the same side of the line as B.

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