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?
Best Answer
There is a discussion of this issue in Dieudonné's History of Functional Analysis, p. 115:
If I may summarize: the idea that one should talk about mathematical objects in terms of the axioms they should satisfy was itself quite new around 1900, and the specific application of this idea to the triangle inequality seems quite likely to have originated with Fréchet for that reason.