The triangle inequality also applies in spherical geometry where the distance between any two points P and Q is the angle they subtend at the center of the sphere, and shortest distances are great circles. As I don't have access to your source, I don't know what it is about their proof that would make it inapplicable to spherical geometry where Pythagoras' theorem doesn't work. If it does apply in spherical geometry, what is it about Staring's proof that doesn't work?