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Pythagoras'theorem doesn't feel sufficient to prove that the diagonal is the the shortest route. It only computes that distance. So is there a way to prove that a diagonal is the shortest route? It seems very intuitive. However, I wanted to see a mathematical proof.

Could the triangle inequality be used?

James Chadwick
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    You are correct in pointing out that the Pythagorean theorem along is not sufficient to prove that "the diagonal" is the shortest route. What you need to prove is that a straight line segment gives the shortest length between two points. (This, of course, depends on what your starting assumptions/axioms are.) This is actually a common intro exercise in calculus of variations, but I'm sure there are more elementary proofs (I haven't thought about this too much so this is why I am writing this as a comment). – Maximal Ideal Apr 11 '23 at 06:33

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