I have found myself in a corner with this proof after working with it for the whole day and would appreciate input. As a reverse to proving that the shortest distance between two points via Pythagoras' theorem, I am attempting to prove the Pythagoras' theorem in an Euclidian space, given that it is known that the shortest distance between two points is a straight line. Is this even possible, or is it possibly even trivial? What am I missing here?
I have tried several different approaches, but none have yet yielded anything. I am thinking the most logical way to go about proving this would be to begin with that the hypotenuse is less or equal than to the sum of the legs of a straight triangle.