We seek the place on the earth’s surface where a vertically suspended bar (or the rings of Saturn) look the biggest.
A machine to solve this problem consists of a mass fixed to the earth’s surface and a couple of bars attached to the mass and constrained to pass through the ends of the bar. A spring between the bars pushes them apart.
The machine reaches equilibrium when the bars are as far apart as possible, hence finding the solution to Regiomontanus’ Problem.
You can drag the ends of the bar to change the problem.
Press Show to see the circumcircle of the ends of the bar and the mass. You should notice something about the position of this circumcircle at equilibrium.