Find symmetric points with respect to the unit circle

Question

I have struggled with an exercise, namely: find the set of symmetric points with respect to the unit circle of a circle given by this equation: $ |z-1|=1$, I have an idea of what this might be. Since points, A and B, which are the intersection points of both circles will not change when taking symmetry and also point 2 will go to 1/2. So I guess that the set will be the line crossing all 3 points A, B and 1/2, but I would like some explanation added to this. Thanks, any help appreciated.

Answer 1

Hint: Let $z$ and $w$ be on circle $|z-1|=1$, then $$|z-1|=1~~~~;~~~~|w-1|=1$$ these points are symmetric respct to the unit circle, means there is a $\theta\in\mathbb R$ such that $$\dfrac{z+w}{2}=e^{i\theta}$$ or $|z+w|=2$, therefore with deleting $w$ among equations $$|z-1|=1~~~~;~~~~|w-1|=1~~~~;~~~~|z+w|=2$$ we can find desired points $z$.