Triangle inequality
Overview
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the triangle inequality states that for any triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side (and the inequality is strict if the triangle is non-degenerate
Degeneracy (mathematics)
In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class....
).
In Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
and some other geometries the triangle inequality is a theorem about distances. In Euclidean geometry, for right triangle
Right triangle
A right triangle or right-angled triangle is a triangle in which one angle is a right angle . The relation between the sides and angles of a right triangle is the basis for trigonometry.-Terminology:The side opposite the right angle is called the hypotenuse...
s it is a consequence of Pythagoras' theorem, and for general triangles a consequence of the law of cosines
Law of cosines
In trigonometry, the law of cosines relates the lengths of the sides of a plane triangle to the cosine of one of its angles. Using notation as in Fig...
, although it may be proven without these theorems. The inequality can be viewed intuitively in either R2 or R3.
