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 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 defining property of norms and measures of distance

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

a relationship between absolute values

where the length of the third side has been replaced by the length of the vector sum u + v. When u and v are real numbers, they can be viewed as vectors in R 1 {\displaystyle \mathbb {R} ^{1}} , and the triangle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 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):

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.

a mathematical result known as triangle inequality

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

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 law of triangles which states that, for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side

Triangle inequality is a law of triangles which states that, for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.

a fundamental concept in geometry

The triangle inequality is a fundamental concept in geometry, and geometry has a number of applications in mathematics and physics.

three inequalities that are true simultaneously

The triangle inequality is three inequalities that are true simultaneously.

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).

This important property of a triangle

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

the relationship between the three sides of a triangle

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

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 proved without these theorems.

a property of absolute values

This is just what's called a triangular inequality; this is just a property of absolute values.

an important theorem that has a number of applications in everyday life

The triangle inequality is an important theorem that has a number of applications in everyday life.