Locus of complex numbers involving arguments

Question

I am trying to find the complex numbers that satisfy

$$ \arg(z-1)=\arg(z^2)$$

I am considering a circle centred at $(1,0),$ but am unsure as to how to proceed.

Answer 1

Well, $$z-1=z^2 \Rightarrow \text{arg }(z-1)=\text{arg }(z^2)$$

So, certainly the points on the circle $z^2=z-1$ (centered at $(1,0)$ with radius one) satisfy the condition.

Also, to include, are the points on the ray along the positive $x-$axis with $x>1$.

We have the locus of points $A$ satisfying the equality as:

$$A=\{z|z-1=z^2 \}\cup\{{z=x|x>1}\}$$