There are plenty of proofs for the statement, but I do not find one which is not rely on other assumptions. Here are some common proofs of this statement: https://en.m.wikipedia.org/wiki/Great_Circle#Derivation_of_shortest_paths This proof require the path to be differentiable, which is not a part of the statement. https://math.stackexchange.com/questions/1180923/shortest-path-on-a-sphere There are several proofs in the page, they are rely on the uniqueness of the shortest path. They thought the uniqueness is intuitive, but I cannot persuade myself on this assumption. Can anyone provide me a strict proof without other assumption of the statement?