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:
 * $\alpha + \beta + \gamma = 0$
 * not all of $\alpha, \beta, \gamma$ are zero.
 * not all of $\alpha, \beta, \gamma$ are zero.

Sufficient Condition
Let $z_1, z_2, z_3$ be collinear.

Then by Condition for Collinearity of Points in Complex Plane: Formulation 1 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 $\paren {b - 1} + 1 + \paren {-b} = 0$.

Necessary Condition
Let $\alpha + \beta + \gamma = 0$ such that:
 * $\alpha z_1 + \beta z_2 + \gamma z_3 = 0$
 * at least one of $\alpha, \beta, \gamma$ is not zero.

let $\alpha \ne 0$.

Then it follows that as $\alpha + \beta + \gamma = 0$, at least one of $\beta$ and $\gamma$ is also non-zero.

let $\beta \ne 0$.

In the following it is immaterial whether $\gamma = 0$ or not.

We have:

Thus it is seen that:
 * $z_2 - z_1 = b \paren {z_3 - z_1}$

for some $b \in \R$.

Hence by Condition for Collinearity of Points in Complex Plane: Formulation 1, $z_1$, $z_2$ and $z_3$ are collinear.