Finding the centre of a circle using arg((z)/(z-2)) = π/3

[phospho]

$$arg(\dfrac{z}{z-2}) = \dfrac{\pi}{3}$$

sketch the locus of z and find the centre of the circle

I've sketched the locus of z but I can't seem to find the centre of the circle. Is there a way to do it algebraically? I've attempted to use z = x + iy, but to no avail.

 

[mfb]

Can you find some (at least 3) points on the circle? The points with minimal/maximal real or imaginary part would be ideal.

A geometric interpretation (for all points, what is the apparent angle between 0 and 2?) is nice, too.

And if you see the symmetry of the problem, it is even better.