triangle inequality
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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.
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