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Common Definition 1: If $A$, $B$, and $C$ are sides of a triangle, then $|A+B| \leq |A| + |B|$. Intuition: We are able to provide an upper bound for the third side of the triangle which is really only met in the degenerate case (when the area of the triangle is 0). Still don't really get it? (so that's why I'm asking)

Common Definition 2: If $x, y, z \in \mathbb{R}$, then $|x-y| \leq |x-z| + |z-y|$

Intuition: For any triangle, the length of any side cannot exceed the sum of the lengths of the other two sides.

Neel Sandell
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  • Are you looking for, "If $A = x - z$ and $B = z - y$, then $A + B = (x - z) + (z - y) = (x-y)$"? Or is the question to do with triangles in some setting, such as a metric space or Euclidean geometry? – Andrew D. Hwang Feb 08 '22 at 02:57
  • I'm not familiar with Euclidean geometry or metric space. I just found these definitions online. – Neel Sandell Feb 08 '22 at 03:27
  • Thank you for the answer however. I didn't realize that we could transform definition 2 to definition 1 by doing that substitution. – Neel Sandell Feb 08 '22 at 03:29
  • Note that it's potentially misleading to call the triangle inequality a "definition." Better might be a property (of various types of geometry) or an axiom (for a metric space). <> Is there a specific question you have about the triangle inequality, or a set of specific circumstances that lead you to ask about it at Math.SE? Either might clarify what type of answer would help. – Andrew D. Hwang Feb 08 '22 at 17:25
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    I appreciate that distinction. I can see how it's a property derived from an observation. The reason I asked about it on here was because I wasn't sure how the two definitions related to each other even though they were called the same thing. – Neel Sandell Feb 09 '22 at 22:27

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