I graphed the Ford circles for the first few terms in the Fibonacci sequence $\frac{1}{1}, \frac{2}{1}, \frac{3}{2}, \ldots$ as well as a circle with radius $\frac{\sqrt{5}}{2}$ about the point $(\frac{1}{2}, 0)$.
Interestingly, the large circle overlaps the Ford circles right where they overlap.
It's one of those things that looks reasonable but I can't really explain. If I didn't know $\phi$ in advance, could I have found that circle?