Condition for Collinearity of Points in Complex Plane/Formulation 2

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

Then:
 * $z_1, z_2, z_3$ are collinear in the complex plane


 * $\exists \alpha, \beta, \gamma \in \R: \alpha z_1 + \beta z_2 + \gamma z_3 = 0$ where $\alpha + \beta + \gamma = 0$.
 * $\exists \alpha, \beta, \gamma \in \R: \alpha z_1 + \beta z_2 + \gamma z_3 = 0$ where $\alpha + \beta + \gamma = 0$.

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

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

Then:

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