Mathematics · Euclidean Geometry

Triangle Inequality Theorem

Reference entry · last updated September 13, 2026

Triangle inequality theorem states that for any non-degenerate triangle, the sum of the lengths of any two sides must be strictly greater than the length of the third side [1].

1. First principles and mathematical formulation

In Euclidean plane geometry, a straight line path represents the shortest distance between two points. If $A$, $B$, and $C$ are three distinct non-collinear vertices, traveling directly from $A$ to $B$ is strictly shorter than traveling from $A$ to $C$ and then from $C$ to $B$ [1].

Let the side lengths opposite vertices $A$, $B$, and $C$ be $a$, $b$, and $c$. A valid non-degenerate triangle exists if and only if all three inequalities hold simultaneously:

$$a + b > c$$

$$a + c > b$$

$$b + c > a$$

If three points are collinear with $C$ lying on segment $AB$, the relation yields $AC + CB = AB$. This forms a degenerate triangle of zero area [2].

2. Existence test for side lengths

To verify whether three given lengths $\{a, b, c\}$ form a triangle, sorting the lengths simplifies the check. Let the sides be ordered such that:

$$a \le b \le c$$

Because $c \ge b \ge a$, the conditions $a + c > b$ and $b + c > a$ are automatically satisfied for positive numbers. Therefore, verification requires only checking that the sum of the two shorter sides exceeds the longest side:

$$a + b > c$$

For example, side lengths $3, 4, 8$ cannot form a triangle because $3 + 4 = 7 \not> 8$. Side lengths $5, 6, 9$ form a triangle because $5 + 6 = 11 > 9$.

3. Finding the range of an unknown side

When two side lengths $a$ and $b$ are known, the third side $x$ is bounded from both below and above by combining the three inequalities:

$$|a - b| < x < a + b$$

The lower bound $|a - b|$ derives from rearranging $a + x > b$ and $b + x > a$. The upper bound $a + b$ derives directly from $a + b > x$. The third side must lie strictly within this open interval.

4. Generalization to vectors and metric spaces

The geometric theorem extends as an axiom across analysis and linear algebra [3]:

Normed vector spaces: For any vectors $\mathbf{u}, \mathbf{v} \in V$:
$$\|\mathbf{u} + \mathbf{v}\| \le \|\mathbf{u}\| + \|\mathbf{v}\|$$
Metric spaces: A metric space $(M, d)$ requires distance function $d$ to satisfy the triangle inequality for all points $x, y, z \in M$:
$$d(x, z) \le d(x, y) + d(y, z)$$
Reverse triangle inequality: For real numbers and normed spaces:
$$|\|\mathbf{u}\| - \|\mathbf{v}\|| \le \|\mathbf{u} - \mathbf{v}\|$$

References

^Euclid, Elements, Book I, Proposition 20.
^H.S.M. Coxeter, Introduction to Geometry, 2nd ed., John Wiley & Sons, 1969.
^Walter Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, 1976.