Three random legs on a round table all 90 degrees apart

Three legs are attached to a round tabletop at three points chosen independently and uniformly around its edge. The table counts as acceptable only if every pair of legs is separated by an angle of at least $90^\circ$. What is the probability that the table is acceptable?

$\tfrac1{16}$. The condition says all three arcs between consecutive legs exceed a quarter of the circle. For $n$ uniform points on a circle, all gaps exceed a fraction $a$ with probability $(1-na)^{n-1}$; with $n=3$, $a=\tfrac14$ this is $(\tfrac14)^2=\tfrac1{16}$. Integrating over the second leg's position gives the same answer.