Question: Why is the triangle inequality at the heart of so many proofs?
Discussion
In many areas of mathematics, from complex analysis to discrete number theory, and even chaos theory, many theorems and smaller lemmas make use of the triangle inequality as part of, or the foundation of, their proof.
$$|a+b|\leq|a|+|b|$$
The school-level understanding of this inequality is that "the most direct path between two points is the shortest" doesn't quite give enough insight into the disproportionate power of the inequality.
I'm not an expert (self-teaching) but I feel a deeper insight might have something to do with the geometry of multi-dimensional objects which provides sufficient constraint to be useful for proving such identities.
Request: I would prefer answers to be accessible to non-experts who are not familiar with terminology like manifolds, metric spaces and fields.