There exist points $A, B,$ and $C$ on a unit sphere, such that the space distance between each pair of points is $\sqrt{2}.$ An ant can only crawl on the surface of the sphere. What is the shortest possible distance of a trip of which the ant starts from $A,$ and then passes through $B$ and $C,$ then returns to $A.$
I think the answer is $\frac{3\pi}{2}.$ Call the center of the sphere $O.$ Essentially, since the shortest path on a sphere connecting two points is the arc of their great circle, consider a random pair of points, in this case lets say $A$ and $B.$ Since $AB= \sqrt{2},$ and the sphere is a unit sphere, we know $\triangle ABO$ is a 45-45-90 triangle. Therefore, the arc of the great circle between two points will be $\frac{2\pi}{4} = \frac{\pi}{2}.$ Since $AB=BC=AC,$ we may simply triple this distance to get the total shortest possible distance of this round trip, which is $\frac{3\pi}{2}.$
Is this solution correct? Thanks in advance.
Also, if my solution is correct, are there any other ways of doing this problem?