Centre of a circle & complex numbers

AI Thread Summary
The discussion revolves around finding the center of a circle defined by the equation arg(z/(z-2)) = π/3. Participants explore algebraic methods to determine the center, with attempts using z = x + iy proving unhelpful. The need for specific points on the circle, particularly those with minimal or maximal real or imaginary parts, is emphasized. A geometric interpretation of the angle between 0 and 2 is also sought, highlighting the symmetry of the problem. Overall, the thread focuses on both algebraic and geometric approaches to solve the problem.
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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.
 
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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.
 

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