expresses a relationship between absolute values

When and are real numbers, they can be viewed as vectors in , and the triangle inequality expresses a relationship between absolute values.

a requirement upon distance

In a metric space with metric , the triangle inequality is a requirement upon distance: for all , , in .

a defining property of norms and measures of distance

The triangle inequality is a defining property of norms and measures of distance.

inequalities involving the parameters of triangles

In geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain conditions.

a defining property of a metric space

The triangular inequality is a defining property of a metric space, while the stronger ultrametric inequality is a defining property of an ultrametric space.

a consequence of a consequence for right triangles

Since the Pythagorean theorem only applies to right triangles, the triangle inequality is a consequence of a consequence for right triangles.

a theorem about distances

In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and the triangle inequality is written using vectors and vector lengths (norms):

consequence of the Pythagorean theorem

In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although cosines may be proven without these theorems.

determining the best upper estimate on the size of the sum of two numbers

The triangle inequality is useful in mathematical analysis for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers.

vectors in two dimensions

The triangle inequality is usually expressed algebraically using vectors in two dimensions.

three inequalities that are true simultaneously

The triangle inequality is three inequalities that are true simultaneously.

In a metric space M with metric d

In a metric space with metric , the triangle inequality is a requirement upon distance: for all , , in .

a mathematical result known as triangle inequality

One way of doing that is by using a mathematical result known as triangle inequality.

pattern recognition technique

a triangle inequality is commonly used as a pattern recognition technique when comparing time series.

In Euclidean geometry for right triangles

In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although cosines may be proven without these theorems.

the relationship between the three sides of a triangle

The triangle inequality theorem describes the relationship between the three sides of a triangle.

the property that guarantees that d(A, B) ≤ d(A, C) + d(B, C)

Then the triangular inequality is the property that guarantees that d(A, B) ≤ d(A, C) + d(B, C).

part of the definition of a metric

In mathematics, triangle inequality is part of the definition of a metric, and distances in mathematics are synonymous to metrics.

This important property of a triangle

This important property of a triangle is known as Triangle inequality.

the fulfilment of the axioms of order for such a relation 'later

The inequality of a triangle means the fulfilment of the axioms of order for such a relation 'later'.