I am currently preparing a talk that revolves around the triangle inequality.
Because this inequality is so well-established, I do not want to (in my talk) belabor too much upon the importance it enjoys. For example, I learned some useful views here. But, these concerns are currently too advanced---for my purposes, I am seeking first some historical background; specifically,
Approximately when, where, and how did the concept of a triangle inequality get formalized, and its importance recognized?
EDIT It seems that the above question is not precise or clear enough. How about the slightly clarified question:
When was it realized (was it Fréchet's 1906 paper cited in the comments?) that the triangle inequality should be a fundamental axiom for defining distances?