Theorem 2.5

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

ax₁ + by₁ = c

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

C = (1 - t)A + tB

= (1 - t)(x₁, y₁) + t(x₂, y₂)

= ((1 - t)x₁ + tx₂, (1 - t)y₁ + ty₂)

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.