Triangle inequality information

Three examples of the triangle inequality for triangles with sides of lengths $x$ , $y$ , $z$ . The top example shows a case where $z$ is much less than the sum $x + y$ of the other two sides, and the bottom example shows a case where the side $z$ is only slightly less than $x + y$ .

In mathematics, the triangle inequality 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.^[1]^[2] This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality.^[3] If $x$ , $y$ , and $z$ are the lengths of the sides of the triangle, with no side being greater than $z$ , then the triangle inequality states that

z\leq x+y,

with equality only in the degenerate case of a triangle with zero area.

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

\|\mathbf {x} +\mathbf {y} \|\leq \|\mathbf {x} \|+\|\mathbf {y} \|,

where the length $z$ of the third side has been replaced by the vector sum $x + y$ . When $x$ and $y$ are real numbers, they can be viewed as vectors in $\mathbb {R} ^{1}$ , and the triangle inequality expresses a relationship between absolute values.

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 it may be proved without these theorems. The inequality can be viewed intuitively in either $\mathbb {R} ^{2}$ or $\mathbb {R} ^{3}$ . The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a $180°$ angle and two $0°$ angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.

In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in $[0, π]$ ) with those endpoints.^[4]^[5]

The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the L^p spaces ( $p \geq 1$ ), and inner product spaces.

^ Wolfram MathWorld – http://mathworld.wolfram.com/TriangleInequality.html
^ Mohamed A. Khamsi; William A. Kirk (2001). "§1.4 The triangle inequality in $R n$ ". An introduction to metric spaces and fixed point theory. Wiley-IEEE. ISBN 0-471-41825-0.
^ for instance, Jacobs, Harold R. (1974), Geometry, W. H. Freeman & Co., p. 246, ISBN 0-7167-0456-0
^ Oliver Brock; Jeff Trinkle; Fabio Ramos (2009). Robotics: Science and Systems IV. MIT Press. p. 195. ISBN 978-0-262-51309-8.
^ Arlan Ramsay; Robert D. Richtmyer (1995). Introduction to hyperbolic geometry. Springer. p. 17. ISBN 0-387-94339-0.

[1] Wolfram MathWorld – http://mathworld.wolfram.com/TriangleInequality.html

[Khamsi-2] Mohamed A. Khamsi; William A. Kirk (2001). "§1.4 The triangle inequality in $R n$ ". An introduction to metric spaces and fixed point theory. Wiley-IEEE. ISBN 0-471-41825-0.

[3] r instance, Jacobs, Harold R. (1974), Geometry, W. H. Freeman & Co., p. 246, ISBN 0-7167-0456-0

[Ramos-4] Oliver Brock; Jeff Trinkle; Fabio Ramos (2009). Robotics: Science and Systems IV. MIT Press. p. 195. ISBN 978-0-262-51309-8.

[Ramsay-5] Arlan Ramsay; Robert D. Richtmyer (1995). Introduction to hyperbolic geometry. Springer. p. 17. ISBN 0-387-94339-0.

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