Theorem 2.5: Let A be a point on the line determined by the equation ax + by = c, and let B be a point not on that line. Then the points on the line determined by A and B which are on the same side of A as B are on the same side of the line ax + by = c as B, and the points on the other side of A from B on the line determined by A and B are on the other side of the line ax + by = c.
Proof: Let A = (x1, y1), and let B = (x2, y2) indicate the coordinates of A and B.
Since A is on the line ax + by = c, we have
Let C be any point on the line determined by A and B. Then by Theorem 2.1, there is a real number t such that
If we want to see on which side of the line ax + by = c this point is, we substitute its coordinates into the left side of the equation determining the line.
Remove parentheses.
Rearrange the terms.
Factor.
Substitute ax1 + by1 = c
If t is positive, then the point will be on the same side of the line ax + by = c as B, and if t is negative, then it will be on the other side.