Stability of a randomly legged round table

Three legs are attached to the rim of a circular tabletop, each at an independent, uniformly random position. What is the probability the table stands up?

$1/4$. The table stands exactly when the three points are not contained in any semicircle. For each point, the chance the other two lie in the half-circle clockwise from it is $\tfrac14$, and these three events are disjoint, so a semicircle contains all three with probability $\tfrac34$.