= Solution
Consider an isolated <intersection> of the two curves. For two distinct <lines>, solving their <linear equations> by elimination gives coordinates in $F$ whenever there is an <intersection>.
For a <line> and a <circle>, solve the <line> for one coordinate with a nonzero <coefficient> and substitute into the <circle> equation. The resulting quadratic has <coefficients> in $F$. A real <root> lies in $F$ if its <discriminant> is a square in $F$, and otherwise in $F(\sqrt D)$ for its positive nonsquare <discriminant> $D\in F$. The other coordinate is an $F$-linear expression in that <root>. Thus both coordinates lie in the same real <quadratic extension>. A <tangent> <intersection> has <discriminant> zero and already lies in $F$.
Write two <circle> equations in normalized form $x^2+y^2+u_ix+v_iy+w_i=0$. Subtracting eliminates the quadratic terms. If the <circles> are distinct and meet, this gives a nontrivial <line>, reducing the problem to the preceding case. Concentric distinct <circles> do not intersect. Therefore
$$
\boxed{\text{each isolated intersection has coordinates in }F\text{ or in }F(\sqrt D),\quad [F(\sqrt D):F]=2.}
$$
The isolated-intersection qualification is implicit in the construction operation. If coincident curves were allowed and every point on their common curve counted as an <intersection> point, the printed assertion would be false: a rational <line> contains points with transcendental coordinates.
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