Complex numbers straight line proof 
Prove that the three distinct points $z_1,z_2$, and $z_3$ lie on the
  same straight line iff $z_3 - z_2 = c(z_2 - z_1)$ for some real number
  $c$ and $z$ is complex.

I know that two vectors are parallel iff one is a scalar multiple of the other, thus $z$ is parallel to $w$ iff $z = cw$. So, from that, does that mean $z_3 - z_2 = c(z_2 - z_1)$ are parallel thus making it lie on the same line?

WLOG let $z_1$ be at the origin, and $z_2 -z_1 = r_1e^{i\theta_1},$ an arbitrary vector from the origin. $z_2$ is the endpoint of this vector, and we have a third point $z_3$ and want to know if the angle of $z_3-z_2 = r_2e^{i\theta_2}$ is that of $z_2-z_1.$
But $$\frac{z_3 -z_2}{z_2-z_1}=\frac{r_1e^{i\theta_1}}{r_2e^{i\theta_2}}= \frac{r_1}{r_2}e^{i(\theta_1-\theta_2)} = c. $$ Since c is real, $\theta_1 -\theta_2$ must equal zero, so they are equal.  
