Probability a Three-Legged Disk Table Stands

You have a perfectly round disk. You attach three legs to the disk at three points chosen independently and uniformly at random on the circumference. The legs are perpendicular to the disk and firmly attached. You flip the disk over so it rests on the three legs. The table stands upright if and only if the center of the disk lies inside the triangle formed by the three leg positions. Otherwise, the table tips over. What is the probability that the table stands?