Inspired by this Sangaku-style question about a constellation of circles, I've come up with the following question.
How can we construct, with straightedge and compass, the following diagram?
Description: In a (large) circle, two equal length chords share a point on the circle. In each of the two segments thus formed, a largest possible circle is inscribed. A third circle touches the large circle and both chords. The three inscribed circles are of equall radii.
The ratio of the large circle's radii to the smaller circles' radii is $\phi+1$, where $\phi=\frac{1+\sqrt5}{2}$ is the golden ratio (proof).
If we construct the chords, then the circles are easy to draw. But how can we construct the chords?
The ratio of the smaller circle to the larger circle is $$\frac{1}{\phi+1}=\frac{3-\sqrt{5}}{2}$$
If you decide on the radius of the larger circle first, say $1$ unit, then draw a straight line of length $3$ units.
Take a point $2$ units from one end of the line, and construct a perpendicular from there of length $1$ unit to make a right-angled triangle whose hypotenuse is $\sqrt{5}$. Mark off the length $\sqrt{5}$ from one end of the first line. Bisect the remaining length of that line to make a length of $$\frac{3-\sqrt{5}}{2}$$
Now that you have the smaller radius you can draw the upper circle first, then the chords, and finally the two side circles, a process which should be fairly routine.