Without using the distance formula, prove that the points A(1, 4), B(3, –2) and C(–3, 16) are collinear.

To prove that the points A(1, 4), B(3, –2), and C(–3, 16) are collinear we must show that the slope between any pair of points is the same.

By formula,

Slope (m) = $\dfrac{y$2 - y1}{x2 - x1}

Substituting values we get,

$\Rightarrow \text{Slope of AB} = \dfrac{-2 - 4}{3 - 1} \\[1em] = \dfrac{-6}{2} \\[1em] = -3. \\[1em] \Rightarrow \text{Slope of BC} = \dfrac{16 -(-2)}{-3 - 3} \\[1em] = \dfrac{16 + 2}{-6} \\[1em] = \dfrac{18}{-6} \\[1em] = -3.$

Slope of AB = Slope of BC

Hence, proved points A, B and C are collinear.