Three annulus intersection problem

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Recently I faced a problem about intersection of three annuli.
Imagine having three annuli same dimensions and you put them next to each other into triangular shape like putting together three circles. Now let's say these annuli can slide over each other and form intersections. The question is what is the maximum ratio between annulus' thickness $(R - r)$ and its radius (we have three same annuli), so that when equally sliding them one over each other from starting triangular position, we do not come across triple intersection, but we have to slide them far enough so that the outer circle of each annulus touches the inner circle of its neighbour's annuli.