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bahamagreen said:"Consider that the shortest distance between two points on the Earth's surface is not a straight line, but by a curved line."
I would consider that it is a line through the interior of the surface. If you stipulate that the line must curve with the sphere's surface, that is fine, but that is a longer distance.
Does this mean I won't be satisfied with a geometric answer?
Well, if you use an analogy half-way and then cheat on it, yeah, you will be dissatisfied. You either accept that the surface analogy REQUIRES you to ignore the "through the interior" or you reject the analogy. You can't have it both ways, you can only try to do so and be frustrated.