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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?

Frank Conry
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    If the two points are neither identical nor antipodal, then the triangle with the two points and the centre of the sphere is not degenerate, and so it lies on a unique plane. This plane's intersection with the sphere then defines a unique circle, which is a great circle as the plane contains the centre of the sphere. Any great circle through the two points would have lie on a plane containing the two points and the centre of the sphere and so the triangle, so would be the same plane and same great circle. – Henry Nov 21 '23 at 16:19
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    In particular, note that the claim fails when the two points are antipodal (that is, are directly opposite each other through the center of the sphere). All great circles passing through a point on the sphere will also pass through its antipodal point, so a pair of antipodal points do not determine a unique great circle. Henry's construction fails in this case, as the two antipodal points and the center of the sphere are collinear, so they do not determine a unique plane, unlike any three non-collinear points. – Paul Sinclair Nov 22 '23 at 16:18

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