Condition for Collinearity of Points in Complex Plane

Theorem
Let $z_1, z_2, z_3$ be distinct complex numbers.

Let $z_1, z_2, z_3$ be collinear in the complex plane.

Then:
 * $\alpha z_1 + \beta z_2 + \gamma z_3 = 0$

where $\alpha, \beta, \gamma$ are real numbers such that $\alpha + \beta + \gamma = 0$.

Proof
Let $z_1, z_2, z_3$ be collinear.

Then there exists a real number $b$ such that:
 * $z_2 - z_1 = b \left({z_3 - z_1}\right)$

Then:

Setting $\alpha = b-1, \beta = 1, \gamma = -b$ fits the bill, as $\left({b - 1}\right) + 1 + \left({-b}\right) = 0$.