I had a debate with a buddy about this. He said you could get a chord by drawing the triangle formed by the two points and the center of the sphere and that chord corresponds to a single great circle arc. I can get to the chord but then projecting that chord on to a path on the surface of the sphere seems non obvious.
So is it true that two points not colinear with the spheres center on the surface of a sphere have a unique great circle connecting them and if so how does one prove it?