There are three circles with different radii arranged such that the smallest circle is contained entirely within the next larger circle and that circle is entirely contained within the largest circle. All three circles are touching at a single point. If the inner circle is rotated, how many rotations of that circle will be required before the points on each of the circles at which they originally touched are again a single point? Assume the radii are such that the points will eventually be perfectly aligned.
The real world application of this, which may help to visualize the problem, is with a grown up version of Spirograph that has multiple rings such that you can put a gear inside a ring inside a ring. The outer ring stays fixed while the inner ring and the gear inside the inner ring rotate.
The simpler question of a gear inside a ring, or a circle inside a circle, has already been asked and answered. For the gear problem the size of the rings and gears are given in number of teeth on the gear or ring and this gives an equation of LCM(G,R)/G, where LCM is least common multiple, G is teeth on the gear and R is teeth on the ring.
The question of gear inside ring inside ring (circle in circle in circle), however, does not seem to have come up here. I know from experimentation that the answer is not a simple extension of the gear in ring equation. Any assistance on this will be greatly appreciated by me and I'm sure many others using this gear set. For extra credit and extreme gratitude, how would the answer be extended to include an additional outer ring (gear in ring in ring in ring)?
We have a small gear with radius $G$, a gear ring (annulus) with inner radius $R_i$ and outer radius $R_o$, and containing circle with radius $C$. One orbit of the ring around the circle will bring it back to its initial position with $\frac{C}{R_o}$ rotations. The small gear will have completed $\frac{C}{R_o}\times\frac{R_i}{G}$ rotations. In order to have the gear and the ring return to their starting position in their starting orientation, there need to be $O$ orbits such that $O\times\frac{C}{R_o}$ and $O\times\frac{C}{R_o}\times\frac{R_i}{G}$ are both integers. $$O=\frac{G\times R_o}{\text{gcd}(G\times R_o,C\times R_i)}$$ Example: $G=21,R_i=28,R_o=35,C=50\implies O=21$
$21$ orbits gives $30$ rotations of the ring and $40$ rotations of the gear. I'm not certain that this will match your observations, but it's the best I can come up with.