Editing Triangle inequality (section)

{{Short description|Property of geometry, also used to generalize the notion of "distance" in metric spaces}}
{{about|the basic inequality <math>c\le a+b</math><!-- DO NOT USE {{math}}. IT DOES NOT WORK INSIDE HATNOTES. -->|other inequalities associated with triangles|List of triangle inequalities}}

[[File:TriangleInequality.svg|thumb|Three examples of the triangle inequality for triangles with sides of lengths {{mvar|x}}, {{mvar|y}}, {{mvar|z}}. The top example shows a case where {{mvar|z}} is much less than the sum {{math|''x'' + ''y''}} of the other two sides,  and the bottom example shows a case where the side {{mvar|z}} is only slightly less than {{math|''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.<ref>Wolfram MathWorld – http://mathworld.wolfram.com/TriangleInequality.html</ref><ref name=Khamsi>
{{cite book |title=An introduction to metric spaces and fixed point theory |author1=Mohamed A. Khamsi |author2=William A. Kirk |chapter-url=https://books.google.com/books?id=4qXbEpAK5eUC&pg=PA8 |chapter=§1.4 The triangle inequality in {{math|'''R'''<sup>n</sup>}} |isbn=0-471-41825-0 |year=2001 |publisher=Wiley-IEEE}}</ref> This statement permits the inclusion of [[Degeneracy (mathematics)#Triangle|degenerate triangles]], but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality.<ref>for instance, {{citation|first=Harold R.|last=Jacobs|title=Geometry|year=1974|publisher=W. H. Freeman & Co.|isbn=0-7167-0456-0|page=246}}</ref> If {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} are the lengths of the sides of a triangle then the triangle inequality states that
:<math>c \leq a + b ,</math>
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 vectors and vector lengths ([[Norm (mathematics)|norms]]):
:<math>\|\mathbf u + \mathbf v\| \leq \|\mathbf u\| + \|\mathbf v\| ,</math>
where the length of the third side has been replaced by the length of the vector sum {{math|'''u''' + '''v'''}}. When {{math|'''u'''}} and {{math|'''v'''}} are [[real number]]s, they can be viewed as vectors in <math>\R^1</math>, and the triangle inequality expresses a relationship between [[absolute value]]s.

In Euclidean geometry, for [[right triangle]]s 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 <math>\R^2</math> or <math>\R^3</math>. 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 {{math|180°}} angle and two {{math|0°}} angles, making the three [[Vertex (geometry)|vertices]] [[Straight line|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 {{math|{{closed-closed|0, ''π''}}}}) with those endpoints.<ref name= Ramos>
{{cite book |title=Robotics: Science and Systems IV |author1=Oliver Brock |author2=Jeff Trinkle |author3=Fabio Ramos |url=https://books.google.com/books?id=fvCaQfBQ7qEC&pg=PA195 |page=195 |isbn=978-0-262-51309-8 |publisher=MIT Press |year=2009}}
</ref><ref name=Ramsay>
{{cite book |title=Introduction to hyperbolic geometry |author1=Arlan Ramsay |author2=Robert D. Richtmyer |url=https://archive.org/details/introductiontohy0000rams |url-access=registration |page=[https://archive.org/details/introductiontohy0000rams/page/17 17] |isbn=0-387-94339-0 |year=1995 |publisher=Springer}}
</ref>

The triangle inequality is a ''defining property'' of [[norm (mathematics)|norms]] and measures of [[Metric (mathematics)#Definition|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 number]]s, [[Euclidean space]]s, the [[Lp space|L<sup>p</sup> space]]s ({{math|''p'' ≥ 1}}), and [[inner product space]]s.