Theorem 2.7: Given points   A   and   B   and a line whose equation is   ax + by = c,   where   A   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).

The equation of the line is   ax + by = c.   We can assume that

because if we find the inequality going the other way, the proof will be similar. 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,   0 < t < 1.

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

But since   0 < t < 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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