= Solution
Put $v=r\omega$, $\beta=v/c<1$ and $\gamma=(1-\beta^2)^{-1/2}$. For a <photon> emitted in laboratory direction $\mathbf n$ toward the stationary receiver, its <frequency> there is
$$
\nu=\frac{\nu_0}{\gamma(1-\mathbf n\cdot\mathbf v_e/c)}.
$$
Since the receiver lies outside the circle, two emission points have a line of sight tangent to the circle. At those points the source <velocity> is directly toward or away from the receiver, so $\mathbf n\cdot\mathbf v_e=\pm v$. Thus the <doppler extrema for a circularly moving source> are
$$
\boxed{\nu_{\max}=\nu_0\sqrt{\frac{1+\beta}{1-\beta}},\qquad \nu_{\min}=\nu_0\sqrt{\frac{1-\beta}{1+\beta}}.}
$$
They do not depend on $d$, provided $d>r$; changing $d$ changes the emission and reception phases of the extrema. Closest and farthest source positions instead have transverse <velocity> relative to that ray, and are not the extrema.
\Image[/past-exam-of-the-relativity-course-of-the-university-of-cambridge/2016/paper-1-rotating-doppler.png]
{title=Original tangent-ray geometry for the largest approaching and receding shifts of a circular source}
For the corotating receiver, <velocities> must be evaluated at the actual emission and reception events. An unobstructed <vacuum> ray satisfies $\mathbf x_r-\mathbf x_e=c\Delta t\,\mathbf n$, while rigid steady rotation gives $\mathbf v_r-\mathbf v_e=\boldsymbol\omega\times(\mathbf x_r-\mathbf x_e)$. Therefore $\mathbf n\cdot\mathbf v_r=\mathbf n\cdot\mathbf v_e$. The emitter and receiver have equal <speed> magnitudes, hence equal <Lorentz factors>. The <frequency ratio of corotating clocks> is consequently
$$
\frac{\nu_r}{\nu_0}=\frac{\gamma_r(1-\mathbf n\cdot\mathbf v_r/c)}{\gamma_e(1-\mathbf n\cdot\mathbf v_e/c)}=\boxed{1}.
$$
Thus \b[the rim receiver measures $\nu_0$], independently of its angular separation. Different transit times or a <Sagnac effect> do not by themselves imply a <frequency> change between these identical-radius clocks. The model treats $\nu_0$ as a fixed proper clock <frequency> and assumes a signal path is available; it does not specify the disc's electromagnetic opacity.
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