So, I just did some crunching, and it turns out that is NOT a true statement, so clearly I'm on the wrong track from the beginning. (Use x=1, y=2, z=0 for a counterexample.)
You're problem here was the confusion of what the actual metric is. Let then to show that the triangle quality hold here you must show that . You, probably used to using the triangle inequality for the usual metric on , automatically thought you must mess with the 's.