Triangle inequality
Our editors will review what you’ve submitted and determine whether to revise the article.
Join Britannica's Publishing Partner Program and our community of experts to gain a global audience for your work!Triangle inequality, in Euclidean geometry, theorem that the sum of any two sides of a triangle is greater than or equal to the third side; in symbols, a + b ≥ c. In essence, the theorem states that the shortest distance between two points is a straight line.
The triangle inequality has counterparts for other metric spaces, or spaces that contain a means of measuring distances. Measures are called norms, which are typically indicated by enclosing an entity from the space in a pair of single or double vertical lines,   or  . For example, real numbers a and b, with the absolute value as a norm, obey a version of the triangle inequality given by a + b ≥ a + b. A vector space given a norm, such as the Euclidean norm (the square root of the sum of the squares of the vector’s components), obeys a version of the triangle inequality for vectors x and y given by x + y ≥ x + y.
With appropriate norms, the triangle inequality holds for complex numbers, integrals, and other abstract spaces in functional analysis.
Learn More in these related Britannica articles:

metric space…these properties is called the triangle inequality. The French mathematician Maurice Fréchet initiated the study of metric spaces in 1905.…

Euclidean geometry
Euclidean geometry , the study of plane and solid figures on the basis of axioms and theorems employed by the Greek mathematician Euclid (c. 300bce ). In its rough outline, Euclidean geometry is the plane and solid geometry commonly taught in secondary schools. Indeed, until the second half of the 19th…