= Solution
Induct through the finitely many construction steps. Initially all available coordinates lie in $\mathbb Q$. If coordinates obtained so far lie in a real <field> $F$, the <line> through two of its points has <coefficients> in $F$. A <circle> with <centre> in $F^2$ and <radius> a <distance> between two such points has equation
$$
(x-a)^2+(y-b)^2=r^2,
$$
whose <coefficients> lie in $F$, since the squared <distance> $r^2$ lies in $F$; the <radius> itself need not lie there.
Part (b) therefore puts every new <intersection> in $F$ or a real <quadratic extension> of $F$. If a step produces several points, adjoin their coordinates successively, omitting extensions that add nothing. A finite construction ends in a real <field> $K$ containing the desired coordinates, with a tower of genuine quadratic extensions
$$
\mathbb Q=F_0\subset F_1\subset\cdots\subset F_n=K.
$$
The <tower law for field extensions> yields
$$
\boxed{[K:\mathbb Q]=\prod_{i=0}^{n-1}[F_{i+1}:F_i]=2^n.}
$$
A construction involving no nontrivial extension has $n=0$ and <degree> one.
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