Cardinality of the set of points on a circle whose coordinates are constructable is $\aleph_0$

Question

Let $C=${$(x,y)\in\mathbb R^{2}|(x-a)^{2}+(y-b)^{2}=r^{2}$} be a circle of radius $r>1$ and centre $(a,b)$ where $a,b,r$ are all constructible. We want to show that the cardinality of the set of points on $C$ whose coordinates are constructible is $\aleph_0$ I do not have a good idea on how to approach this question. Is it enough that I show any coordinate with constructable numbers on the 2d plane is $\aleph_0$? Or should I show a bijection between the natural numbers and the set of points on C who's coordinates are constructable? If so, what bijection should I use? And I also do not know what is the use of the information that $a,b,r$ are constructable and $r<1$

Answer 1

I would follow the first approach, to show that the set of constructible points is $\aleph_0$. At any stage in a construction, there are only finitely many points that can be constructed. The number of stages of a construction is also finite. The set of constructible points is then a countable union of finite sets.