# Conditions on angles between three points on a sphere (which are uniformly distributed)

Question: Let $A,B,C$ be three random, uniformly distributed, independant points on a sphere. What is the probability that none of these three points is at an angle superior than $\pi/2$ from the two others.

As of right now, I know that wlog, we can assume that $A$ is on the north pole and that $B$ is on a great circle passing through the north and south pole. Then, I am trying to calculate the area which isn't covered by the northern hemisphere or by the hemisphere centered at $B$. I have trouble figuring out how to deal clearly with all of these cases.

Thank you in advance for you help!