Let's say $z_1, z_2, z_3$ are three complex numbers in AP. Does that mean they lie on a circle in the complex plane?
So, $z_2 = \frac{z_1 + z_3}{2}$
$z_2$ is the midpoint of $z_1$ and $z_3$. It means they are collinear, so they lie on a line.
Is that information enough?
What if $z_2$ is the midpoint of $z_1$ and $z_3$ and they lie on an arc?
This is my doubt!