---
tag: 01M6CW9VM6
kind: statement
role: axiom
name: Triangle inequality for a metric
aliases: [triangle-inequality-for-a-metric]
msc: ["54E35"]
uses: [063W5JMY85]
sources:
  - url: https://en.wikipedia.org/wiki/Triangle_inequality
    locator: Metric space
  - url: https://en.wikipedia.org/wiki/Metric_space
    locator: Definition
---

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]].
For all points $x, y, z \in X$,
$$d(x, z) \le d(x, y) + d(y, z).$$

## Remarks

This is one of the four axioms of a metric.
For $X = \mathbb{R}$ with $d(x, y) = |x - y|$, it is the [[triangle inequality|03HF5PMY5T]] for real numbers, with $a = x - y$ and $b = y - z$.
A consequence is the reverse form $|d(x, y) - d(x, z)| \le d(y, z)$ for all $x, y, z \in X$.

## Explanations

### L1 human

A direct path from $x$ to $z$ is never longer than a path that goes through a third point $y$.
In the plane, this says that one side of a triangle is at most the sum of the other two sides.