Triangle inequality

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class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style> <div role="note" class="hatnote navigation-not-searchable"> This article is about the basic inequality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\leq a+b}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> c </mi> <mo> ≤ </mo> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle c\leq a+b} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af0192f95f340bfc5ba28442bec7763893b4f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.173ex; height:2.343ex;" alt="{\displaystyle c\leq a+b}">. For other inequalities associated with triangles, see <a href="https://en-m-wikipedia-org.translate.goog/wiki/List_of_triangle_inequalities?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="List of triangle inequalities">List of triangle inequalities</a>. </div> In <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematics">mathematics</a>, the triangle inequality states that for any <a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Triangle">triangle</a>, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-1">[1]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Khamsi-2">[2]</a> This statement permits the inclusion of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Degeneracy_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Triangle" title="Degeneracy (mathematics)">degenerate triangles</a>, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-3">[3]</a> If a, b, and c are the lengths of the sides of a triangle then the triangle inequality states that <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:TriangleInequality.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/TriangleInequality.svg/220px-TriangleInequality.svg.png" decoding="async" width="220" height="212" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/b/b2/TriangleInequality.svg/330px-TriangleInequality.svg.png 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/b/b2/TriangleInequality.svg/440px-TriangleInequality.svg.png 2x" data-file-width="938" data-file-height="906"></a> <figcaption> 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. </figcaption> </figure> <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\leq a+b,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> c </mi> <mo> ≤ </mo> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle c\leq a+b,} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22ff9e0576b17c5f9495758847133d25a21b0cb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.82ex; height:2.509ex;" alt="{\displaystyle c\leq a+b,}"> </dd> </dl> with equality only in the degenerate case of a triangle with zero area. In <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euclidean_geometry?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euclidean geometry">Euclidean geometry</a> and some other geometries, the triangle inequality is a theorem about vectors and vector lengths (<a href="https://en-m-wikipedia-org.translate.goog/wiki/Norm_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Norm (mathematics)">norms</a>): <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {u} +\mathbf {v} \|\leq \|\mathbf {u} \|+\|\mathbf {v} \|,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> <mo> ≤ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> <mo> + </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|\mathbf {u} +\mathbf {v} \|\leq \|\mathbf {u} \|+\|\mathbf {v} \|,} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a85e31ebcb4131c320fee1290408fe786a7ffd86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.193ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {u} +\mathbf {v} \|\leq \|\mathbf {u} \|+\|\mathbf {v} \|,}"> </dd> </dl> where the length of the third side has been replaced by the length of the vector sum u + v. When u and v are <a href="https://en-m-wikipedia-org.translate.goog/wiki/Real_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Real number">real numbers</a>, they can be viewed as vectors in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{1}} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec2e282911e406fc800fb1095093667d66f18c7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{1}}">, and the triangle inequality expresses a relationship between <a href="https://en-m-wikipedia-org.translate.goog/wiki/Absolute_value?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Absolute value">absolute values</a>. In Euclidean geometry, for <a href="https://en-m-wikipedia-org.translate.goog/wiki/Right_triangle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Right triangle">right triangles</a> the triangle inequality is a consequence of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Pythagorean_theorem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Pythagorean theorem">Pythagorean theorem</a>, and for general triangles, a consequence of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Law_of_cosines?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Law of cosines">law of cosines</a>, although it may be proved without these theorems. The inequality can be viewed intuitively in either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{2}} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{3}} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Vertex_(geometry)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vertex (geometry)">vertices</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Straight_line?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Straight line">collinear</a>, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line. In <a href="https://en-m-wikipedia-org.translate.goog/wiki/Spherical_geometry?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Spherical geometry">spherical geometry</a>, the shortest distance between two points is an arc of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Great_circle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Great circle">great circle</a>, 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.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Ramos-4">[4]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Ramsay-5">[5]</a> The triangle inequality is a defining property of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Norm_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Norm (mathematics)">norms</a> and measures of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Metric_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Definition" class="mw-redirect" title="Metric (mathematics)">distance</a>. 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 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Real_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Real number">real numbers</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euclidean_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euclidean space">Euclidean spaces</a>, the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Lp_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lp space">Lp spaces</a> (p ≥ 1), and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Inner_product_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Inner product space">inner product spaces</a>. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="en" dir="ltr"> <h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Euclidean_geometry">1 Euclidean geometry</a> <ul> <li class="toclevel-2 tocsection-2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Mathematical_expression_of_the_constraint_on_the_sides_of_a_triangle">1.1 Mathematical expression of the constraint on the sides of a triangle</a></li> <li class="toclevel-2 tocsection-3"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Right_triangle">1.2 Right triangle</a></li> <li class="toclevel-2 tocsection-4"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Examples_of_use">1.3 Examples of use</a></li> <li class="toclevel-2 tocsection-5"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Generalization_to_any_polygon">1.4 Generalization to any polygon</a> <ul> <li class="toclevel-3 tocsection-6"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Example_of_the_generalized_polygon_inequality_for_a_quadrilateral">1.4.1 Example of the generalized polygon inequality for a quadrilateral</a></li> <li class="toclevel-3 tocsection-7"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Relationship_with_shortest_paths">1.4.2 Relationship with shortest paths</a></li> </ul></li> <li class="toclevel-2 tocsection-8"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Converse">1.5 Converse</a></li> <li class="toclevel-2 tocsection-9"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Generalization_to_higher_dimensions">1.6 Generalization to higher dimensions</a></li> </ul></li> <li class="toclevel-1 tocsection-10"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Normed_vector_space">2 Normed vector space</a> <ul> <li class="toclevel-2 tocsection-11"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Example_norms">2.1 Example norms</a></li> </ul></li> <li class="toclevel-1 tocsection-12"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Metric_space">3 Metric space</a></li> <li class="toclevel-1 tocsection-13"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Reverse_triangle_inequality">4 Reverse triangle inequality</a></li> <li class="toclevel-1 tocsection-14"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Triangle_inequality_for_cosine_similarity">5 Triangle inequality for cosine similarity</a></li> <li class="toclevel-1 tocsection-15"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Reversal_in_Minkowski_space">6 Reversal in Minkowski space</a></li> <li class="toclevel-1 tocsection-16"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#See_also">7 See also</a></li> <li class="toclevel-1 tocsection-17"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Notes">8 Notes</a></li> <li class="toclevel-1 tocsection-18"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#References">9 References</a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <h2 id="Euclidean_geometry">Euclidean geometry</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Euclidean geometry" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Euclid_triangle_inequality.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Euclid_triangle_inequality.svg/220px-Euclid_triangle_inequality.svg.png" decoding="async" width="220" height="171" class="mw-file-element" data-file-width="535" data-file-height="417"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 171px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Euclid_triangle_inequality.svg/220px-Euclid_triangle_inequality.svg.png" data-width="220" data-height="171" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Euclid_triangle_inequality.svg/330px-Euclid_triangle_inequality.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Euclid_triangle_inequality.svg/440px-Euclid_triangle_inequality.svg.png 2x" data-class="mw-file-element"> </a> <figcaption> Euclid's construction for proof of the triangle inequality for plane geometry. </figcaption> </figure> Euclid proved the triangle inequality for distances in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euclidean_geometry?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euclidean geometry">plane geometry</a> using the construction in the figure.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Jacobs-6">[6]</a> Beginning with triangle ABC, an isosceles triangle is constructed with one side taken as BC and the other equal leg BD along the extension of side AB. It then is argued that angle β has larger measure than angle α, so side AD is longer than side AC. However: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AD}}={\overline {AB}}+{\overline {BD}}={\overline {AB}}+{\overline {BC}},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi> A </mi> <mi> D </mi> </mrow> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi> A </mi> <mi> B </mi> </mrow> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi> B </mi> <mi> D </mi> </mrow> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi> A </mi> <mi> B </mi> </mrow> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi> B </mi> <mi> C </mi> </mrow> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {AD}}={\overline {AB}}+{\overline {BD}}={\overline {AB}}+{\overline {BC}},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f69ecac3d78525eef1dcbd7593cecf46d7c703a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.068ex; height:3.343ex;" alt="{\displaystyle {\overline {AD}}={\overline {AB}}+{\overline {BD}}={\overline {AB}}+{\overline {BC}},}"> </noscript><span class="lazy-image-placeholder" style="width: 31.068ex;height: 3.343ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f69ecac3d78525eef1dcbd7593cecf46d7c703a" data-alt="{\displaystyle {\overline {AD}}={\overline {AB}}+{\overline {BD}}={\overline {AB}}+{\overline {BC}},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> so the sum of the lengths of sides AB and BC is larger than the length of AC. This proof appears in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euclid%27s_Elements?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euclid's Elements">Euclid's Elements</a>, Book 1, Proposition 20.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Joyce-7">[7]</a> <div class="mw-heading mw-heading3"> <h3 id="Mathematical_expression_of_the_constraint_on_the_sides_of_a_triangle">Mathematical expression of the constraint on the sides of a triangle</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Mathematical expression of the constraint on the sides of a triangle" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths a, b, c that are all positive and excludes the degenerate case of zero area): <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b>c,\quad b+c>a,\quad c+a>b.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> > </mo> <mi> c </mi> <mo> , </mo> <mspace width="1em"></mspace> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> > </mo> <mi> a </mi> <mo> , </mo> <mspace width="1em"></mspace> <mi> c </mi> <mo> + </mo> <mi> a </mi> <mo> > </mo> <mi> b </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a+b>c,\quad b+c>a,\quad c+a>b.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109c1c77f5f1a29d0af753083dd25d2f3ff6ddd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.879ex; height:2.509ex;" alt="{\displaystyle a+b>c,\quad b+c>a,\quad c+a>b.}"> </noscript><span class="lazy-image-placeholder" style="width: 34.879ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109c1c77f5f1a29d0af753083dd25d2f3ff6ddd4" data-alt="{\displaystyle a+b>c,\quad b+c>a,\quad c+a>b.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> A more succinct form of this inequality system can be shown to be <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a-b|<c<a+b.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> a </mi> <mo> − </mo> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> < </mo> <mi> c </mi> <mo> < </mo> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle |a-b|<c<a+b.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2798e693d2f3eb2fffcbaebc74d88abd47b29f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.28ex; height:2.843ex;" alt="{\displaystyle |a-b|<c<a+b.}"> </noscript><span class="lazy-image-placeholder" style="width: 19.28ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2798e693d2f3eb2fffcbaebc74d88abd47b29f21" data-alt="{\displaystyle |a-b|<c<a+b.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Another way to state it is <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(a,b,c)<a+b+c-\max(a,b,c)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix"> max </mo> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> <mo stretchy="false"> ) </mo> <mo> < </mo> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> − </mo> <mo movablelimits="true" form="prefix"> max </mo> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \max(a,b,c)<a+b+c-\max(a,b,c)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ad3af7ae686c35fbf4160dbac6852d7043df02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.728ex; height:2.843ex;" alt="{\displaystyle \max(a,b,c)<a+b+c-\max(a,b,c)}"> </noscript><span class="lazy-image-placeholder" style="width: 37.728ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ad3af7ae686c35fbf4160dbac6852d7043df02" data-alt="{\displaystyle \max(a,b,c)<a+b+c-\max(a,b,c)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> implying <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\max(a,b,c)<a+b+c}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 2 </mn> <mo movablelimits="true" form="prefix"> max </mo> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> <mo stretchy="false"> ) </mo> <mo> < </mo> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 2\max(a,b,c)<a+b+c} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0c0ea2697cb84c590e32c0d2777af66f9c68355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25ex; height:2.843ex;" alt="{\displaystyle 2\max(a,b,c)<a+b+c}"> </noscript><span class="lazy-image-placeholder" style="width: 25ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0c0ea2697cb84c590e32c0d2777af66f9c68355" data-alt="{\displaystyle 2\max(a,b,c)<a+b+c}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> and thus that the longest side length is less than the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Semiperimeter?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Semiperimeter">semiperimeter</a>. A mathematically equivalent formulation is that the area of a triangle with sides a, b, c must be a real number greater than zero. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Heron%27s_formula?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Heron's formula">Heron's formula</a> for the area is <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}4\cdot {\text{area}}&={\sqrt {(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}\\&={\sqrt {-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2a^{2}c^{2}+2b^{2}c^{2}}}.\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn> 4 </mn> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> area </mtext> </mrow> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mo> − </mo> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo> − </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> − </mo> <mi> c </mi> <mo stretchy="false"> ) </mo> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo> − </mo> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mo> − </mo> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mo> − </mo> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msup> <mo> + </mo> <mn> 2 </mn> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mn> 2 </mn> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mn> 2 </mn> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}4\cdot {\text{area}}&={\sqrt {(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}\\&={\sqrt {-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2a^{2}c^{2}+2b^{2}c^{2}}}.\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69860e3cdf566c6756124896a953270f196891ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:57.989ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}4\cdot {\text{area}}&={\sqrt {(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}\\&={\sqrt {-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2a^{2}c^{2}+2b^{2}c^{2}}}.\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 57.989ex;height: 8.843ex;vertical-align: -3.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69860e3cdf566c6756124896a953270f196891ab" data-alt="{\displaystyle {\begin{aligned}4\cdot {\text{area}}&={\sqrt {(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}\\&={\sqrt {-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2a^{2}c^{2}+2b^{2}c^{2}}}.\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero). The triangle inequality provides two more interesting constraints for triangles whose sides are a, b, c, where a ≥ b ≥ c and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> ϕ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \phi } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"> </noscript><span class="lazy-image-placeholder" style="width: 1.385ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" data-alt="{\displaystyle \phi }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Golden_ratio?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Golden ratio">golden ratio</a>, as <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1<{\frac {a+c}{b}}<3}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 1 </mn> <mo> < </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> c </mi> </mrow> <mi> b </mi> </mfrac> </mrow> <mo> < </mo> <mn> 3 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 1<{\frac {a+c}{b}}<3} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1aec38a22e6e0b075c17df3fc6859708e8e7bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.435ex; height:5.176ex;" alt="{\displaystyle 1<{\frac {a+c}{b}}<3}"> </noscript><span class="lazy-image-placeholder" style="width: 14.435ex;height: 5.176ex;vertical-align: -2.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1aec38a22e6e0b075c17df3fc6859708e8e7bab" data-alt="{\displaystyle 1<{\frac {a+c}{b}}<3}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq \min \left({\frac {a}{b}},{\frac {b}{c}}\right)<\phi .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 1 </mn> <mo> ≤ </mo> <mo movablelimits="true" form="prefix"> min </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> a </mi> <mi> b </mi> </mfrac> </mrow> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> b </mi> <mi> c </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> < </mo> <mi> ϕ </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 1\leq \min \left({\frac {a}{b}},{\frac {b}{c}}\right)<\phi .} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2d1fd3cea600eab036f3570c0ecb93a65c53e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.018ex; height:6.176ex;" alt="{\displaystyle 1\leq \min \left({\frac {a}{b}},{\frac {b}{c}}\right)<\phi .}"> </noscript><span class="lazy-image-placeholder" style="width: 22.018ex;height: 6.176ex;vertical-align: -2.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2d1fd3cea600eab036f3570c0ecb93a65c53e5" data-alt="{\displaystyle 1\leq \min \left({\frac {a}{b}},{\frac {b}{c}}\right)<\phi .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-8">[8]</a> </dd> </dl> <div class="mw-heading mw-heading3"> <h3 id="Right_triangle">Right triangle</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Right triangle" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Isosceles_triangle_made_of_right_triangles.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Isosceles_triangle_made_of_right_triangles.svg/220px-Isosceles_triangle_made_of_right_triangles.svg.png" decoding="async" width="220" height="219" class="mw-file-element" data-file-width="545" data-file-height="543"> </noscript><span class="lazy-image-placeholder" style="width: 220px;height: 219px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Isosceles_triangle_made_of_right_triangles.svg/220px-Isosceles_triangle_made_of_right_triangles.svg.png" data-width="220" data-height="219" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Isosceles_triangle_made_of_right_triangles.svg/330px-Isosceles_triangle_made_of_right_triangles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Isosceles_triangle_made_of_right_triangles.svg/440px-Isosceles_triangle_made_of_right_triangles.svg.png 2x" data-class="mw-file-element"> </a> <figcaption> Isosceles triangle with equal sides AB = AC divided into two right triangles by an altitude drawn from one of the two base angles. </figcaption> </figure> In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Palmer-9">[9]</a> The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle ADC. An isosceles triangle ABC is constructed with equal sides AB = AC. From the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_postulate?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Triangle postulate">triangle postulate</a>, the angles in the right triangle ADC satisfy: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +\gamma =\pi /2\ .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> α </mi> <mo> + </mo> <mi> γ </mi> <mo> = </mo> <mi> π </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mtext>   </mtext> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \alpha +\gamma =\pi /2\ .} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b098f381a35a729477ee7d43adb41b891d804a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.573ex; height:2.843ex;" alt="{\displaystyle \alpha +\gamma =\pi /2\ .}"> </noscript><span class="lazy-image-placeholder" style="width: 13.573ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b098f381a35a729477ee7d43adb41b891d804a1" data-alt="{\displaystyle \alpha +\gamma =\pi /2\ .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Likewise, in the isosceles triangle ABC, the angles satisfy: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\beta +\gamma =\pi \ .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 2 </mn> <mi> β </mi> <mo> + </mo> <mi> γ </mi> <mo> = </mo> <mi> π </mi> <mtext>   </mtext> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 2\beta +\gamma =\pi \ .} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0abfdac2d493d64cd2d74f91082ea78c07774ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.255ex; height:2.676ex;" alt="{\displaystyle 2\beta +\gamma =\pi \ .}"> </noscript><span class="lazy-image-placeholder" style="width: 12.255ex;height: 2.676ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0abfdac2d493d64cd2d74f91082ea78c07774ef" data-alt="{\displaystyle 2\beta +\gamma =\pi \ .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Therefore, <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\pi /2-\gamma ,\ \mathrm {while} \ \beta =\pi /2-\gamma /2\ ,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> α </mi> <mo> = </mo> <mi> π </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mo> − </mo> <mi> γ </mi> <mo> , </mo> <mtext>   </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> w </mi> <mi mathvariant="normal"> h </mi> <mi mathvariant="normal"> i </mi> <mi mathvariant="normal"> l </mi> <mi mathvariant="normal"> e </mi> </mrow> <mtext>   </mtext> <mi> β </mi> <mo> = </mo> <mi> π </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mo> − </mo> <mi> γ </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 2 </mn> <mtext>   </mtext> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \alpha =\pi /2-\gamma ,\ \mathrm {while} \ \beta =\pi /2-\gamma /2\ ,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b82e3182c24e204b132c0067b97ae59d914ae78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.58ex; height:2.843ex;" alt="{\displaystyle \alpha =\pi /2-\gamma ,\ \mathrm {while} \ \beta =\pi /2-\gamma /2\ ,}"> </noscript><span class="lazy-image-placeholder" style="width: 35.58ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b82e3182c24e204b132c0067b97ae59d914ae78" data-alt="{\displaystyle \alpha =\pi /2-\gamma ,\ \mathrm {while} \ \beta =\pi /2-\gamma /2\ ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> and so, in particular, <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <\beta \ .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> α </mi> <mo> < </mo> <mi> β </mi> <mtext>   </mtext> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \alpha <\beta \ .} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6886b24d363352cab4d9c918baa52bd54266ad30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.146ex; height:2.509ex;" alt="{\displaystyle \alpha <\beta \ .}"> </noscript><span class="lazy-image-placeholder" style="width: 7.146ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6886b24d363352cab4d9c918baa52bd54266ad30" data-alt="{\displaystyle \alpha <\beta \ .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> That means side AD, which is opposite to angle α, is shorter than side AB, which is opposite to the larger angle β. But AB = AC. Hence: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AC}}>{\overline {AD}}\ .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi> A </mi> <mi> C </mi> </mrow> <mo accent="false"> ¯ </mo> </mover> </mrow> <mo> > </mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi> A </mi> <mi> D </mi> </mrow> <mo accent="false"> ¯ </mo> </mover> </mrow> <mtext>   </mtext> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\overline {AC}}>{\overline {AD}}\ .} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01e17af6a1cb2948a9f11e401f0ccc35f4605418" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.801ex; height:3.009ex;" alt="{\displaystyle {\overline {AC}}>{\overline {AD}}\ .}"> </noscript><span class="lazy-image-placeholder" style="width: 11.801ex;height: 3.009ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01e17af6a1cb2948a9f11e401f0ccc35f4605418" data-alt="{\displaystyle {\overline {AC}}>{\overline {AD}}\ .}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> A similar construction shows AC > DC, establishing the theorem. An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point B:<a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Zawaira-10">[10]</a> (i) as depicted (which is to be proved), or (ii) B coincident with D (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle γ, which would violate the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_postulate?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Triangle postulate">triangle postulate</a>), or lastly, (iii) B interior to the right triangle between points A and D (in which case angle ABC is an exterior angle of a right triangle BDC and therefore larger than π/2, meaning the other base angle of the isosceles triangle also is greater than π/2 and their sum exceeds π in violation of the triangle postulate). This theorem establishing inequalities is sharpened by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Pythagoras%27_theorem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Pythagoras' theorem">Pythagoras' theorem</a> to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides. <div class="mw-heading mw-heading3"> <h3 id="Examples_of_use">Examples of use</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Examples of use" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> Consider a triangle whose sides are in an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Arithmetic_progression?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Arithmetic progression">arithmetic progression</a> and let the sides be a, a + d, a + 2d. Then the triangle inequality requires that <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcccl}0&<&a&<&2a+3d,\\0&<&a+d&<&2a+2d,\\0&<&a+2d&<&2a+d.\end{array}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mi> a </mi> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mn> 2 </mn> <mi> a </mi> <mo> + </mo> <mn> 3 </mn> <mi> d </mi> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mi> a </mi> <mo> + </mo> <mi> d </mi> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mn> 2 </mn> <mi> a </mi> <mo> + </mo> <mn> 2 </mn> <mi> d </mi> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mi> a </mi> <mo> + </mo> <mn> 2 </mn> <mi> d </mi> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mn> 2 </mn> <mi> a </mi> <mo> + </mo> <mi> d </mi> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{array}{rcccl}0&<&a&<&2a+3d,\\0&<&a+d&<&2a+2d,\\0&<&a+2d&<&2a+d.\end{array}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391314cbecc93ed607fc4b8dee1737c6ac7c9193" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:29.527ex; height:9.176ex;" alt="{\displaystyle {\begin{array}{rcccl}0&<&a&<&2a+3d,\\0&<&a+d&<&2a+2d,\\0&<&a+2d&<&2a+d.\end{array}}}"> </noscript><span class="lazy-image-placeholder" style="width: 29.527ex;height: 9.176ex;vertical-align: -4.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391314cbecc93ed607fc4b8dee1737c6ac7c9193" data-alt="{\displaystyle {\begin{array}{rcccl}0&<&a&<&2a+3d,\\0&<&a+d&<&2a+2d,\\0&<&a+2d&<&2a+d.\end{array}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> To satisfy all these inequalities requires <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>0{\text{ and }}-{\frac {a}{3}}<d<a.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> a </mi> <mo> > </mo> <mn> 0 </mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>  and  </mtext> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi> a </mi> <mn> 3 </mn> </mfrac> </mrow> <mo> < </mo> <mi> d </mi> <mo> < </mo> <mi> a </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a>0{\text{ and }}-{\frac {a}{3}}<d<a.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eefd4566b5230891d4f4c840f9eb34b11212a85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.595ex; height:4.676ex;" alt="{\displaystyle a>0{\text{ and }}-{\frac {a}{3}}<d<a.}"> </noscript><span class="lazy-image-placeholder" style="width: 24.595ex;height: 4.676ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eefd4566b5230891d4f4c840f9eb34b11212a85" data-alt="{\displaystyle a>0{\text{ and }}-{\frac {a}{3}}<d<a.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-11">[11]</a> </dd> </dl> When d is chosen such that d = a/3, it generates a right triangle that is always similar to the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Pythagorean_triple?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Pythagorean triple">Pythagorean triple</a> with sides 3, 4, 5. Now consider a triangle whose sides are in a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Geometric_progression?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Geometric progression">geometric progression</a> and let the sides be a, ar, ar2. Then the triangle inequality requires that <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcccl}0&<&a&<&ar+ar^{2},\\0&<&ar&<&a+ar^{2},\\0&<&\!ar^{2}&<&a+ar.\end{array}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mi> a </mi> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mi> a </mi> <mi> r </mi> <mo> + </mo> <mi> a </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mi> a </mi> <mi> r </mi> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mi> a </mi> <mo> + </mo> <mi> a </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mspace width="negativethinmathspace"></mspace> <mi> a </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mi> a </mi> <mo> + </mo> <mi> a </mi> <mi> r </mi> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{array}{rcccl}0&<&a&<&ar+ar^{2},\\0&<&ar&<&a+ar^{2},\\0&<&\!ar^{2}&<&a+ar.\end{array}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b134a9284023d120b636ad3246039596b8bdd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:26.865ex; height:9.509ex;" alt="{\displaystyle {\begin{array}{rcccl}0&<&a&<&ar+ar^{2},\\0&<&ar&<&a+ar^{2},\\0&<&\!ar^{2}&<&a+ar.\end{array}}}"> </noscript><span class="lazy-image-placeholder" style="width: 26.865ex;height: 9.509ex;vertical-align: -4.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88b134a9284023d120b636ad3246039596b8bdd2" data-alt="{\displaystyle {\begin{array}{rcccl}0&<&a&<&ar+ar^{2},\\0&<&ar&<&a+ar^{2},\\0&<&\!ar^{2}&<&a+ar.\end{array}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> The first inequality requires a > 0; consequently it can be divided through and eliminated. With a > 0, the middle inequality only requires r > 0. This now leaves the first and third inequalities needing to satisfy <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}r^{2}+r-1&{}>0\\r^{2}-r-1&{}<0.\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> r </mi> <mo> − </mo> <mn> 1 </mn> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo> > </mo> <mn> 0 </mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <mi> r </mi> <mo> − </mo> <mn> 1 </mn> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo> < </mo> <mn> 0. </mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}r^{2}+r-1&{}>0\\r^{2}-r-1&{}<0.\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b86ade1523d82a7beb060017d3a5bd1fc79acaf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.654ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}r^{2}+r-1&{}>0\\r^{2}-r-1&{}<0.\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 15.654ex;height: 6.176ex;vertical-align: -2.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b86ade1523d82a7beb060017d3a5bd1fc79acaf0" data-alt="{\displaystyle {\begin{aligned}r^{2}+r-1&{}>0\\r^{2}-r-1&{}<0.\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> The first of these quadratic inequalities requires r to range in the region beyond the value of the positive root of the quadratic equation r2 + r − 1 = 0, i.e. r > φ − 1 where φ is the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Golden_ratio?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Golden ratio">golden ratio</a>. The second quadratic inequality requires r to range between 0 and the positive root of the quadratic equation r2 − r − 1 = 0, i.e. 0 < r < φ. The combined requirements result in r being confined to the range <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi -1<r<\varphi \,{\text{ and }}a>0.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> φ </mi> <mo> − </mo> <mn> 1 </mn> <mo> < </mo> <mi> r </mi> <mo> < </mo> <mi> φ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>  and  </mtext> </mrow> <mi> a </mi> <mo> > </mo> <mn> 0. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \varphi -1<r<\varphi \,{\text{ and }}a>0.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e74742556d6dabf681246e5fc50b1a9ae4c58af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.722ex; height:2.676ex;" alt="{\displaystyle \varphi -1<r<\varphi \,{\text{ and }}a>0.}"> </noscript><span class="lazy-image-placeholder" style="width: 25.722ex;height: 2.676ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e74742556d6dabf681246e5fc50b1a9ae4c58af" data-alt="{\displaystyle \varphi -1<r<\varphi \,{\text{ and }}a>0.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-12">[12]</a> </dd> </dl> When r the common ratio is chosen such that r = √φ it generates a right triangle that is always similar to the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Kepler_triangle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kepler triangle">Kepler triangle</a>. <div class="mw-heading mw-heading3"> <h3 id="Generalization_to_any_polygon">Generalization to any polygon</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Generalization to any polygon" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The triangle inequality can be extended by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematical_induction?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematical induction">mathematical induction</a> to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths. <div class="mw-heading mw-heading4"> <h4 id="Example_of_the_generalized_polygon_inequality_for_a_quadrilateral">Example of the generalized polygon inequality for a quadrilateral</h4> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Example of the generalized polygon inequality for a quadrilateral" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> Consider a quadrilateral whose sides are in a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Geometric_progression?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Geometric progression">geometric progression</a> and let the sides be a, ar, ar2, ar3. Then the generalized polygon inequality requires that <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rcccl}0&<&a&<&ar+ar^{2}+ar^{3}\\0&<&ar&<&a+ar^{2}+ar^{3}\\0&<&ar^{2}&<&a+ar+ar^{3}\\0&<&ar^{3}&<&a+ar+ar^{2}.\end{array}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center center center left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mi> a </mi> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mi> a </mi> <mi> r </mi> <mo> + </mo> <mi> a </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> a </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mi> a </mi> <mi> r </mi> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mi> a </mi> <mo> + </mo> <mi> a </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> a </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mi> a </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mi> a </mi> <mo> + </mo> <mi> a </mi> <mi> r </mi> <mo> + </mo> <mi> a </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mi> a </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mtd> <mtd> <mo> < </mo> </mtd> <mtd> <mi> a </mi> <mo> + </mo> <mi> a </mi> <mi> r </mi> <mo> + </mo> <mi> a </mi> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{array}{rcccl}0&<&a&<&ar+ar^{2}+ar^{3}\\0&<&ar&<&a+ar^{2}+ar^{3}\\0&<&ar^{2}&<&a+ar+ar^{3}\\0&<&ar^{3}&<&a+ar+ar^{2}.\end{array}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a0540b2d59823ab14ff014d9feeb0f4924d1e6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; margin-top: -0.225ex; width:32.778ex; height:12.843ex;" alt="{\displaystyle {\begin{array}{rcccl}0&<&a&<&ar+ar^{2}+ar^{3}\\0&<&ar&<&a+ar^{2}+ar^{3}\\0&<&ar^{2}&<&a+ar+ar^{3}\\0&<&ar^{3}&<&a+ar+ar^{2}.\end{array}}}"> </noscript><span class="lazy-image-placeholder" style="width: 32.778ex;height: 12.843ex;vertical-align: -5.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a0540b2d59823ab14ff014d9feeb0f4924d1e6e" data-alt="{\displaystyle {\begin{array}{rcccl}0&<&a&<&ar+ar^{2}+ar^{3}\\0&<&ar&<&a+ar^{2}+ar^{3}\\0&<&ar^{2}&<&a+ar+ar^{3}\\0&<&ar^{3}&<&a+ar+ar^{2}.\end{array}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> These inequalities for a > 0 reduce to the following <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{3}+r^{2}+r-1>0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> r </mi> <mo> − </mo> <mn> 1 </mn> <mo> > </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r^{3}+r^{2}+r-1>0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb6a6872f3f738c8720148a82957e5a52702461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.199ex; height:2.843ex;" alt="{\displaystyle r^{3}+r^{2}+r-1>0}"> </noscript><span class="lazy-image-placeholder" style="width: 19.199ex;height: 2.843ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb6a6872f3f738c8720148a82957e5a52702461" data-alt="{\displaystyle r^{3}+r^{2}+r-1>0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{3}-r^{2}-r-1<0.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <mo> − </mo> <msup> <mi> r </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <mi> r </mi> <mo> − </mo> <mn> 1 </mn> <mo> < </mo> <mn> 0. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle r^{3}-r^{2}-r-1<0.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e15de5fd427d06be5e9d4f7d16b61e5c7e6629b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.846ex; height:2.843ex;" alt="{\displaystyle r^{3}-r^{2}-r-1<0.}"> </noscript><span class="lazy-image-placeholder" style="width: 19.846ex;height: 2.843ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e15de5fd427d06be5e9d4f7d16b61e5c7e6629b" data-alt="{\displaystyle r^{3}-r^{2}-r-1<0.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-13">[13]</a> </dd> </dl> The left-hand side polynomials of these two inequalities have roots that are the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Generalizations_of_Fibonacci_numbers?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Tribonacci_numbers" title="Generalizations of Fibonacci numbers">tribonacci constant</a> and its reciprocal. Consequently, r is limited to the range 1/t < r < t where t is the tribonacci constant. <div class="mw-heading mw-heading4"> <h4 id="Relationship_with_shortest_paths">Relationship with shortest paths</h4> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Relationship with shortest paths" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <figure typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Arclength.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Arclength.svg/300px-Arclength.svg.png" decoding="async" width="300" height="78" class="mw-file-element" data-file-width="582" data-file-height="152"> </noscript><span class="lazy-image-placeholder" style="width: 300px;height: 78px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Arclength.svg/300px-Arclength.svg.png" data-width="300" data-height="78" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Arclength.svg/450px-Arclength.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Arclength.svg/600px-Arclength.svg.png 2x" data-class="mw-file-element"> </a> <figcaption> The arc length of a curve is defined as the least upper bound of the lengths of polygonal approximations. </figcaption> </figure> This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line. No polygonal path between two points is shorter than the line between them. This implies that no curve can have an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Arc_length?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Arc length">arc length</a> less than the distance between its endpoints. By definition, the arc length of a curve is the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Least_upper_bound?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Least upper bound">least upper bound</a> of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-14">[14]</a> <div class="mw-heading mw-heading3"> <h3 id="Converse">Converse</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Converse" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths. In either case, if the side lengths are a, b, c we can attempt to place a triangle in the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euclidean_plane?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euclidean plane">Euclidean plane</a> as shown in the diagram. We need to prove that there exists a real number h consistent with the values a, b, and c, in which case this triangle exists. <figure typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Triangle_with_notations_3.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Triangle_with_notations_3.svg/270px-Triangle_with_notations_3.svg.png" decoding="async" width="270" height="169" class="mw-file-element" data-file-width="320" data-file-height="200"> </noscript><span class="lazy-image-placeholder" style="width: 270px;height: 169px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Triangle_with_notations_3.svg/270px-Triangle_with_notations_3.svg.png" data-width="270" data-height="169" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Triangle_with_notations_3.svg/405px-Triangle_with_notations_3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Triangle_with_notations_3.svg/540px-Triangle_with_notations_3.svg.png 2x" data-class="mw-file-element"> </a> <figcaption> Triangle with altitude h cutting base c into d + (c − d). </figcaption> </figure> By the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Pythagorean_theorem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Pythagorean theorem">Pythagorean theorem</a> we have b2 = h2 + d2 and a2 = h2 + (c − d)2 according to the figure at the right. Subtracting these yields a2 − b2 = c2 − 2cd. This equation allows us to express d in terms of the sides of the triangle: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d={\frac {-a^{2}+b^{2}+c^{2}}{2c}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> d </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo> − </mo> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mrow> <mn> 2 </mn> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle d={\frac {-a^{2}+b^{2}+c^{2}}{2c}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf182a76b8b2d1cd9ffaaddec260a1705bd5287" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.683ex; height:5.676ex;" alt="{\displaystyle d={\frac {-a^{2}+b^{2}+c^{2}}{2c}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 19.683ex;height: 5.676ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf182a76b8b2d1cd9ffaaddec260a1705bd5287" data-alt="{\displaystyle d={\frac {-a^{2}+b^{2}+c^{2}}{2c}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> For the height of the triangle we have that h2 = b2 − d2. By replacing d with the formula given above, we have <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{2}=b^{2}-\left({\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)^{2}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> h </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo> − </mo> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mrow> <mn> 2 </mn> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle h^{2}=b^{2}-\left({\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)^{2}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d4d658c8294ff5e0aed14db782121e4866f9d6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.228ex; height:6.676ex;" alt="{\displaystyle h^{2}=b^{2}-\left({\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)^{2}.}"> </noscript><span class="lazy-image-placeholder" style="width: 30.228ex;height: 6.676ex;vertical-align: -2.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d4d658c8294ff5e0aed14db782121e4866f9d6d" data-alt="{\displaystyle h^{2}=b^{2}-\left({\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)^{2}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> For a real number h to satisfy this, h2 must be non-negative: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}0&\leq b^{2}-\left({\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)^{2}\\[4pt]0&\leq \left(b-{\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)\left(b+{\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)\\[4pt]0&\leq \left(a^{2}-(b-c)^{2})((b+c)^{2}-a^{2}\right)\\[6pt]0&\leq (a+b-c)(a-b+c)(b+c+a)(b+c-a)\\[6pt]0&\leq (a+b-c)(a+c-b)(b+c-a)\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mi></mi> <mo> ≤ </mo> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <msup> <mrow> <mo> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo> − </mo> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mrow> <mn> 2 </mn> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mi></mi> <mo> ≤ </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo> − </mo> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mrow> <mn> 2 </mn> <mi> c </mi> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo> − </mo> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mi> c </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mrow> <mn> 2 </mn> <mi> c </mi> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mi></mi> <mo> ≤ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi> b </mi> <mo> − </mo> <mi> c </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> − </mo> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mi></mi> <mo> ≤ </mo> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> − </mo> <mi> c </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo> − </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mi> a </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> − </mo> <mi> a </mi> <mo stretchy="false"> ) </mo> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mi></mi> <mo> ≤ </mo> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> − </mo> <mi> c </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo> + </mo> <mi> c </mi> <mo> − </mo> <mi> b </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> − </mo> <mi> a </mi> <mo stretchy="false"> ) </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}0&\leq b^{2}-\left({\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)^{2}\\[4pt]0&\leq \left(b-{\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)\left(b+{\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)\\[4pt]0&\leq \left(a^{2}-(b-c)^{2})((b+c)^{2}-a^{2}\right)\\[6pt]0&\leq (a+b-c)(a-b+c)(b+c+a)(b+c-a)\\[6pt]0&\leq (a+b-c)(a+c-b)(b+c-a)\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f266b2a02c0f9643c3c251cd0cc417af539466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.171ex; width:49.362ex; height:27.509ex;" alt="{\displaystyle {\begin{aligned}0&\leq b^{2}-\left({\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)^{2}\\[4pt]0&\leq \left(b-{\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)\left(b+{\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)\\[4pt]0&\leq \left(a^{2}-(b-c)^{2})((b+c)^{2}-a^{2}\right)\\[6pt]0&\leq (a+b-c)(a-b+c)(b+c+a)(b+c-a)\\[6pt]0&\leq (a+b-c)(a+c-b)(b+c-a)\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 49.362ex;height: 27.509ex;vertical-align: -13.171ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f266b2a02c0f9643c3c251cd0cc417af539466" data-alt="{\displaystyle {\begin{aligned}0&\leq b^{2}-\left({\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)^{2}\\[4pt]0&\leq \left(b-{\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)\left(b+{\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)\\[4pt]0&\leq \left(a^{2}-(b-c)^{2})((b+c)^{2}-a^{2}\right)\\[6pt]0&\leq (a+b-c)(a-b+c)(b+c+a)(b+c-a)\\[6pt]0&\leq (a+b-c)(a+c-b)(b+c-a)\end{aligned}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> which holds if the triangle inequality is satisfied for all sides. Therefore, there does exist a real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> h </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle h} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"> </noscript><span class="lazy-image-placeholder" style="width: 1.339ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" data-alt="{\displaystyle h}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  consistent with the sides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> c </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a,b,c} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}"> </noscript><span class="lazy-image-placeholder" style="width: 5.302ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" data-alt="{\displaystyle a,b,c}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , and the triangle exists. If each triangle inequality holds <a href="https://en-m-wikipedia-org.translate.goog/wiki/Strict_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Strict inequality">strictly</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h>0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> h </mi> <mo> > </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle h>0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbddb7a5cca6170575e4e73e769fbb434c2a3d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle h>0}"> </noscript><span class="lazy-image-placeholder" style="width: 5.6ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbddb7a5cca6170575e4e73e769fbb434c2a3d71" data-alt="{\displaystyle h>0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> h </mi> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle h=0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe239e1050529410001cc1c0b3245945bc69709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle h=0}"> </noscript><span class="lazy-image-placeholder" style="width: 5.6ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe239e1050529410001cc1c0b3245945bc69709" data-alt="{\displaystyle h=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , the triangle is degenerate. <div class="mw-heading mw-heading3"> <h3 id="Generalization_to_higher_dimensions">Generalization to higher dimensions</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Generalization to higher dimensions" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style> <table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"> <tbody> <tr> <td class="mbox-text"> <div class="mbox-text-span"> This section does not <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:Citing_sources?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wikipedia:Citing sources">cite</a> any <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:Verifiability?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wikipedia:Verifiability">sources</a>. Please help <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:EditPage/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:EditPage/Triangle inequality">improve this section</a> by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Help:Referencing_for_beginners?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:Verifiability?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (October 2021) (<a href="https://en-m-wikipedia-org.translate.goog/wiki/Help:Maintenance_template_removal?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Help:Maintenance template removal">Learn how and when to remove this message</a>) </div></td> </tr> </tbody> </table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"> <div role="note" class="hatnote navigation-not-searchable"> See also: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Distance_geometry?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Cayley%E2%80%93Menger_determinants" title="Distance geometry">Distance geometry § Cayley–Menger determinants</a> </div> The area of a triangular face of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tetrahedron?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tetrahedron">tetrahedron</a> is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an (n − 1)-<a href="https://en-m-wikipedia-org.translate.goog/wiki/Facet_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Facet (mathematics)">facet</a> of an n-<a href="https://en-m-wikipedia-org.translate.goog/wiki/Simplex?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Simplex">simplex</a> is less than or equal to the sum of the hypervolumes of the other n facets. Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Polytope?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Polytope">polytope</a> of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets. In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points A, B, C, and Z in Euclidean space such that distances <dl> <dd> AB = BC = CA = 26 </dd> </dl> and <dl> <dd> AZ = BZ = CZ = 14. </dd> </dl> However, points with such distances cannot exist: the area of the 26–26–26 equilateral triangle ABC is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 169{\sqrt {3}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn> 169 </mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 3 </mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\textstyle 169{\sqrt {3}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b00f116292ae7a7334438b0067c29b7228721fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.586ex; height:3.009ex;" alt="{\textstyle 169{\sqrt {3}}}"> </noscript><span class="lazy-image-placeholder" style="width: 6.586ex;height: 3.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b00f116292ae7a7334438b0067c29b7228721fa" data-alt="{\textstyle 169{\sqrt {3}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , which is larger than three times <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 39{\sqrt {3}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn> 39 </mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn> 3 </mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\textstyle 39{\sqrt {3}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9408bbf2e8c9550f806b42af2c340c1645155167" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.423ex; height:3.009ex;" alt="{\textstyle 39{\sqrt {3}}}"> </noscript><span class="lazy-image-placeholder" style="width: 5.423ex;height: 3.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9408bbf2e8c9550f806b42af2c340c1645155167" data-alt="{\textstyle 39{\sqrt {3}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , the area of a 26–14–14 isosceles triangle (all by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Heron%27s_formula?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Heron's formula">Heron's formula</a>), and so the arrangement is forbidden by the tetrahedral inequality. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <h2 id="Normed_vector_space">Normed vector space</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Normed vector space" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <figure typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Vector-triangle-inequality.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Vector-triangle-inequality.svg/300px-Vector-triangle-inequality.svg.png" decoding="async" width="300" height="129" class="mw-file-element" data-file-width="625" data-file-height="269"> </noscript><span class="lazy-image-placeholder" style="width: 300px;height: 129px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Vector-triangle-inequality.svg/300px-Vector-triangle-inequality.svg.png" data-width="300" data-height="129" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Vector-triangle-inequality.svg/450px-Vector-triangle-inequality.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Vector-triangle-inequality.svg/600px-Vector-triangle-inequality.svg.png 2x" data-class="mw-file-element"> </a> <figcaption> Triangle inequality for norms of vectors. </figcaption> </figure> In a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Normed_vector_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Normed vector space">normed vector space</a> V, one of the defining properties of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Norm_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Norm (mathematics)">norm</a> is the triangle inequality: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {u} +\mathbf {v} \|\leq \|\mathbf {u} \|+\|\mathbf {v} \|\quad \forall \,\mathbf {u} ,\mathbf {v} \in V}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> <mo> ≤ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> <mo> + </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> <mspace width="1em"></mspace> <mi mathvariant="normal"> ∀ </mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo> ∈ </mo> <mi> V </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|\mathbf {u} +\mathbf {v} \|\leq \|\mathbf {u} \|+\|\mathbf {v} \|\quad \forall \,\mathbf {u} ,\mathbf {v} \in V} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd426bb9f128d5ce04ac7a82121fe69c05cf6c91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.107ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {u} +\mathbf {v} \|\leq \|\mathbf {u} \|+\|\mathbf {v} \|\quad \forall \,\mathbf {u} ,\mathbf {v} \in V}"> </noscript><span class="lazy-image-placeholder" style="width: 34.107ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd426bb9f128d5ce04ac7a82121fe69c05cf6c91" data-alt="{\displaystyle \|\mathbf {u} +\mathbf {v} \|\leq \|\mathbf {u} \|+\|\mathbf {v} \|\quad \forall \,\mathbf {u} ,\mathbf {v} \in V}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> That is, the norm of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Vector_sum?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Addition_and_subtraction" class="mw-redirect" title="Vector sum">sum of two vectors</a> is at most as large as the sum of the norms of the two vectors. This is also referred to as <a href="https://en-m-wikipedia-org.translate.goog/wiki/Subadditivity?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Subadditivity">subadditivity</a>. For any proposed function to behave as a norm, it must satisfy this requirement.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Kress-15">[15]</a> If the normed space is <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euclidean_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euclidean space">Euclidean</a>, or, more generally, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Strictly_convex_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Strictly convex space">strictly convex</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {u} +\mathbf {v} \|=\|\mathbf {u} \|+\|\mathbf {v} \|}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> <mo> = </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> <mo> + </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|\mathbf {u} +\mathbf {v} \|=\|\mathbf {u} \|+\|\mathbf {v} \|} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43db653ff387f716b0e4494847b2c342b66b0011" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.547ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {u} +\mathbf {v} \|=\|\mathbf {u} \|+\|\mathbf {v} \|}"> </noscript><span class="lazy-image-placeholder" style="width: 21.547ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43db653ff387f716b0e4494847b2c342b66b0011" data-alt="{\displaystyle \|\mathbf {u} +\mathbf {v} \|=\|\mathbf {u} \|+\|\mathbf {v} \|}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  if and only if the triangle formed by u, v, and u + v, is degenerate, that is, u and v are on the same ray, i.e., u = 0 or v = 0, or u = α v for some α > 0. This property characterizes strictly convex normed spaces such as the ℓp spaces with 1 < p < ∞. However, there are normed spaces in which this is not true. For instance, consider the plane with the ℓ1 norm (the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Manhattan_distance?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Manhattan distance">Manhattan distance</a>) and denote u = (1, 0) and v = (0, 1). Then the triangle formed by u, v, and u + v, is non-degenerate but <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\mathbf {u} +\mathbf {v} \|=\|(1,1)\|=|1|+|1|=2=\|\mathbf {u} \|+\|\mathbf {v} \|.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> <mo> = </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> = </mo> <mn> 2 </mn> <mo> = </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> <mo> + </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|\mathbf {u} +\mathbf {v} \|=\|(1,1)\|=|1|+|1|=2=\|\mathbf {u} \|+\|\mathbf {v} \|.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1458e7fb856e87c5e525d71e6f4cc31d2bf287d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.897ex; height:2.843ex;" alt="{\displaystyle \|\mathbf {u} +\mathbf {v} \|=\|(1,1)\|=|1|+|1|=2=\|\mathbf {u} \|+\|\mathbf {v} \|.}"> </noscript><span class="lazy-image-placeholder" style="width: 47.897ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1458e7fb856e87c5e525d71e6f4cc31d2bf287d" data-alt="{\displaystyle \|\mathbf {u} +\mathbf {v} \|=\|(1,1)\|=|1|+|1|=2=\|\mathbf {u} \|+\|\mathbf {v} \|.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> <div class="mw-heading mw-heading3"> <h3 id="Example_norms">Example norms</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=11&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Example norms" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The <a href="https://en-m-wikipedia-org.translate.goog/wiki/Absolute_value?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Absolute value">absolute value</a> is a norm for the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Real_line?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Real line">real line</a>; as required, the absolute value satisfies the triangle inequality for any real numbers u and v. If u and v have the same sign or either of them is zero, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |u+v|=|u|+|v|.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle |u+v|=|u|+|v|.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb813e1e40f0248cdf40396744dcf722e0954109" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.222ex; height:2.843ex;" alt="{\displaystyle |u+v|=|u|+|v|.}"> </noscript><span class="lazy-image-placeholder" style="width: 18.222ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb813e1e40f0248cdf40396744dcf722e0954109" data-alt="{\displaystyle |u+v|=|u|+|v|.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  If u and v have opposite signs, then <a href="https://en-m-wikipedia-org.translate.goog/wiki/Without_loss_of_generality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Without loss of generality">without loss of generality</a> assume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |u|>|v|.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> > </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle |u|>|v|.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8be6fb4ccdcc121588dade5b2e0dc9c3b7a9f27a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.79ex; height:2.843ex;" alt="{\displaystyle |u|>|v|.}"> </noscript><span class="lazy-image-placeholder" style="width: 8.79ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8be6fb4ccdcc121588dade5b2e0dc9c3b7a9f27a" data-alt="{\displaystyle |u|>|v|.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |u+v|=|u|-|v|<|u|+|v|.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> < </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle |u+v|=|u|-|v|<|u|+|v|.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc52047e64419f1c9a1c25fc071a04c0e35e655e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.205ex; height:2.843ex;" alt="{\displaystyle |u+v|=|u|-|v|<|u|+|v|.}"> </noscript><span class="lazy-image-placeholder" style="width: 29.205ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc52047e64419f1c9a1c25fc071a04c0e35e655e" data-alt="{\displaystyle |u+v|=|u|-|v|<|u|+|v|.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Combining these cases:<a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Stewart-16">[16]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |u+v|\leq |u|+|v|.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> ≤ </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle |u+v|\leq |u|+|v|.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e7dd5741100aa30dcc531841647c8bc5b03e835" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.222ex; height:2.843ex;" alt="{\displaystyle |u+v|\leq |u|+|v|.}"> </noscript><span class="lazy-image-placeholder" style="width: 18.222ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e7dd5741100aa30dcc531841647c8bc5b03e835" data-alt="{\displaystyle |u+v|\leq |u|+|v|.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The triangle inequality is useful in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematical_analysis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematical analysis">mathematical analysis</a> for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers. There is also a lower estimate, which can be found using the reverse triangle inequality which states that for any real numbers u and v, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |u-v|\geq {\bigl |}|u|-|v|{\bigr |}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mo> − </mo> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> ≥ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> | </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> | </mo> </mrow> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle |u-v|\geq {\bigl |}|u|-|v|{\bigr |}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed7aa87c40ca56d8f1de2c56834812bed69b576" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.515ex; height:3.176ex;" alt="{\displaystyle |u-v|\geq {\bigl |}|u|-|v|{\bigr |}.}"> </noscript><span class="lazy-image-placeholder" style="width: 19.515ex;height: 3.176ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed7aa87c40ca56d8f1de2c56834812bed69b576" data-alt="{\displaystyle |u-v|\geq {\bigl |}|u|-|v|{\bigr |}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  The <a href="https://en-m-wikipedia-org.translate.goog/wiki/Taxicab_norm?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Taxicab norm">taxicab norm</a> or 1-norm is one generalization absolute value to higher dimensions. To find the norm of a vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=(v_{1},v_{2},\ldots v_{n}),}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> <mo> = </mo> <mo stretchy="false"> ( </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v=(v_{1},v_{2},\ldots v_{n}),} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73f988ca596b2b4397c6eb405968baf0e28a660d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.57ex; height:2.843ex;" alt="{\displaystyle v=(v_{1},v_{2},\ldots v_{n}),}"> </noscript><span class="lazy-image-placeholder" style="width: 18.57ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73f988ca596b2b4397c6eb405968baf0e28a660d" data-alt="{\displaystyle v=(v_{1},v_{2},\ldots v_{n}),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  just add the absolute value of each component separately, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|v\|_{1}=|v_{1}|+|v_{2}|+\dotsb +|v_{n}|.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <msub> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> + </mo> <mo> ⋯ </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|v\|_{1}=|v_{1}|+|v_{2}|+\dotsb +|v_{n}|.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90a355195bd470f163c1e67d0179ab72908ae236" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.087ex; height:2.843ex;" alt="{\displaystyle \|v\|_{1}=|v_{1}|+|v_{2}|+\dotsb +|v_{n}|.}"> </noscript><span class="lazy-image-placeholder" style="width: 30.087ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90a355195bd470f163c1e67d0179ab72908ae236" data-alt="{\displaystyle \|v\|_{1}=|v_{1}|+|v_{2}|+\dotsb +|v_{n}|.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The Euclidean norm or 2-norm defines the length of translation vectors in an n-dimensional <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euclidean_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euclidean space">Euclidean space</a> in terms of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Cartesian_coordinate_system?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cartesian coordinate system">Cartesian coordinate system</a>. For a vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=(v_{1},v_{2},\ldots v_{n}),}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> <mo> = </mo> <mo stretchy="false"> ( </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v=(v_{1},v_{2},\ldots v_{n}),} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73f988ca596b2b4397c6eb405968baf0e28a660d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.57ex; height:2.843ex;" alt="{\displaystyle v=(v_{1},v_{2},\ldots v_{n}),}"> </noscript><span class="lazy-image-placeholder" style="width: 18.57ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73f988ca596b2b4397c6eb405968baf0e28a660d" data-alt="{\displaystyle v=(v_{1},v_{2},\ldots v_{n}),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  its length is defined using the n-dimensional <a href="https://en-m-wikipedia-org.translate.goog/wiki/Pythagorean_theorem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Pythagorean theorem">Pythagorean theorem</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|v\|_{2}={\sqrt {|v_{1}|^{2}+|v_{2}|^{2}+\dotsb +|v_{n}|^{2}}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <msub> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo> ⋯ </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|v\|_{2}={\sqrt {|v_{1}|^{2}+|v_{2}|^{2}+\dotsb +|v_{n}|^{2}}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e6abb20f1d86d7bd5fd6de864c5ec50ac49b765" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:35.574ex; height:4.843ex;" alt="{\displaystyle \|v\|_{2}={\sqrt {|v_{1}|^{2}+|v_{2}|^{2}+\dotsb +|v_{n}|^{2}}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 35.574ex;height: 4.843ex;vertical-align: -1.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e6abb20f1d86d7bd5fd6de864c5ec50ac49b765" data-alt="{\displaystyle \|v\|_{2}={\sqrt {|v_{1}|^{2}+|v_{2}|^{2}+\dotsb +|v_{n}|^{2}}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The inner product is norm in any <a href="https://en-m-wikipedia-org.translate.goog/wiki/Inner_product_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Inner product space">inner product space</a>, a generalization of Euclidean vector spaces including infinite-dimensional examples. The triangle inequality follows from the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Cauchy%E2%80%93Schwarz_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cauchy–Schwarz inequality">Cauchy–Schwarz inequality</a> as follows: Given vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> u </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" data-alt="{\displaystyle u}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> </noscript><span class="lazy-image-placeholder" style="width: 1.128ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" data-alt="{\displaystyle v}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , and denoting the inner product as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle u,v\rangle }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨ </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ⟩ </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle u,v\rangle } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6704456ffbbed3155bc5dd40e03459129011c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.301ex; height:2.843ex;" alt="{\displaystyle \langle u,v\rangle }"> </noscript><span class="lazy-image-placeholder" style="width: 5.301ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6704456ffbbed3155bc5dd40e03459129011c99" data-alt="{\displaystyle \langle u,v\rangle }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> :<a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Stillwell-17">[17]</a> <dl> <dd> <table> <tbody> <tr> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u+v\|^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> + </mo> <mi> v </mi> <msup> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|u+v\|^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bce9dee15b5f8e5ad78da6ff0173c02653e748e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.677ex; height:3.176ex;" alt="{\displaystyle \|u+v\|^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 8.677ex;height: 3.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bce9dee15b5f8e5ad78da6ff0173c02653e748e2" data-alt="{\displaystyle \|u+v\|^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\langle u+v,u+v\rangle }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> = </mo> <mo fence="false" stretchy="false"> ⟨ </mo> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mo> , </mo> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ⟩ </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle =\langle u+v,u+v\rangle } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05ffa49ff92e16f2c652380cb7fedd4a399bdd1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.892ex; height:2.843ex;" alt="{\displaystyle =\langle u+v,u+v\rangle }"> </noscript><span class="lazy-image-placeholder" style="width: 15.892ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05ffa49ff92e16f2c652380cb7fedd4a399bdd1e" data-alt="{\displaystyle =\langle u+v,u+v\rangle }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </td> </tr> <tr> <td></td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\|u\|^{2}+\langle u,v\rangle +\langle v,u\rangle +\|v\|^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> = </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <msup> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mo fence="false" stretchy="false"> ⟨ </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ⟩ </mo> <mo> + </mo> <mo fence="false" stretchy="false"> ⟨ </mo> <mi> v </mi> <mo> , </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ⟩ </mo> <mo> + </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <msup> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle =\|u\|^{2}+\langle u,v\rangle +\langle v,u\rangle +\|v\|^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2705065d27ea5b0fd218d85303b066d060dc656" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.791ex; height:3.176ex;" alt="{\displaystyle =\|u\|^{2}+\langle u,v\rangle +\langle v,u\rangle +\|v\|^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 30.791ex;height: 3.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2705065d27ea5b0fd218d85303b066d060dc656" data-alt="{\displaystyle =\|u\|^{2}+\langle u,v\rangle +\langle v,u\rangle +\|v\|^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </td> </tr> <tr> <td></td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq \|u\|^{2}+2|\langle u,v\rangle |+\|v\|^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> ≤ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <msup> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo fence="false" stretchy="false"> ⟨ </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ⟩ </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> + </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <msup> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \leq \|u\|^{2}+2|\langle u,v\rangle |+\|v\|^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/794527280cf331b5ad64fc534ffffd4dc4e583c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.106ex; height:3.176ex;" alt="{\displaystyle \leq \|u\|^{2}+2|\langle u,v\rangle |+\|v\|^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 25.106ex;height: 3.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/794527280cf331b5ad64fc534ffffd4dc4e583c8" data-alt="{\displaystyle \leq \|u\|^{2}+2|\langle u,v\rangle |+\|v\|^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </td> </tr> <tr> <td></td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq \|u\|^{2}+2\|u\|\|v\|+\|v\|^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> ≤ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <msup> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mn> 2 </mn> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> + </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <msup> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \leq \|u\|^{2}+2\|u\|\|v\|+\|v\|^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7848d5ac90707f1df8903810e2012d12eb95e6f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.619ex; height:3.176ex;" alt="{\displaystyle \leq \|u\|^{2}+2\|u\|\|v\|+\|v\|^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 25.619ex;height: 3.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7848d5ac90707f1df8903810e2012d12eb95e6f4" data-alt="{\displaystyle \leq \|u\|^{2}+2\|u\|\|v\|+\|v\|^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (by the Cauchy–Schwarz inequality)</td> </tr> <tr> <td></td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\left(\|u\|+\|v\|\right)^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> = </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> + </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> </mrow> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle =\left(\|u\|+\|v\|\right)^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82224a4b73b926096aa7664e56d0b3f6b286bc5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.264ex; height:3.343ex;" alt="{\displaystyle =\left(\|u\|+\|v\|\right)^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 15.264ex;height: 3.343ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82224a4b73b926096aa7664e56d0b3f6b286bc5d" data-alt="{\displaystyle =\left(\|u\|+\|v\|\right)^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</td> </tr> </tbody> </table> </dd> </dl> The Cauchy–Schwarz inequality turns into an equality if and only if u and v are linearly dependent. The inequality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle u,v\rangle +\langle v,u\rangle \leq 2\left|\left\langle u,v\right\rangle \right|}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨ </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ⟩ </mo> <mo> + </mo> <mo fence="false" stretchy="false"> ⟨ </mo> <mi> v </mi> <mo> , </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ⟩ </mo> <mo> ≤ </mo> <mn> 2 </mn> <mrow> <mo> | </mo> <mrow> <mo> ⟨ </mo> <mrow> <mi> u </mi> <mo> , </mo> <mi> v </mi> </mrow> <mo> ⟩ </mo> </mrow> <mo> | </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle u,v\rangle +\langle v,u\rangle \leq 2\left|\left\langle u,v\right\rangle \right|} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d30cb307abe6c44389e560738e49b4ab827dca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.684ex; height:2.843ex;" alt="{\displaystyle \langle u,v\rangle +\langle v,u\rangle \leq 2\left|\left\langle u,v\right\rangle \right|}"> </noscript><span class="lazy-image-placeholder" style="width: 24.684ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d30cb307abe6c44389e560738e49b4ab827dca" data-alt="{\displaystyle \langle u,v\rangle +\langle v,u\rangle \leq 2\left|\left\langle u,v\right\rangle \right|}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  turns into an equality for linearly dependent <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> u </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" data-alt="{\displaystyle u}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> </noscript><span class="lazy-image-placeholder" style="width: 1.128ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" data-alt="{\displaystyle v}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  if and only if one of the vectors u or v is a nonnegative scalar of the other. Taking the square root of the final result gives the triangle inequality. The <a href="https://en-m-wikipedia-org.translate.goog/wiki/P-norm?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="P-norm">p-norm</a> is a generalization of taxicab and Euclidean norms, using an arbitrary positive integer exponent, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|v\|_{p}={\bigl (}|v_{1}|^{p}+|v_{2}|^{p}+\dotsb +|v_{n}|^{p}{\bigr )}^{1/p},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <msub> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> p </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> p </mi> </mrow> </msup> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> p </mi> </mrow> </msup> <mo> + </mo> <mo> ⋯ </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> p </mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mi> p </mi> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|v\|_{p}={\bigl (}|v_{1}|^{p}+|v_{2}|^{p}+\dotsb +|v_{n}|^{p}{\bigr )}^{1/p},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac874518f031c598337c9acad39373e62f903141" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.103ex; height:3.843ex;" alt="{\displaystyle \|v\|_{p}={\bigl (}|v_{1}|^{p}+|v_{2}|^{p}+\dotsb +|v_{n}|^{p}{\bigr )}^{1/p},}"> </noscript><span class="lazy-image-placeholder" style="width: 38.103ex;height: 3.843ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac874518f031c598337c9acad39373e62f903141" data-alt="{\displaystyle \|v\|_{p}={\bigl (}|v_{1}|^{p}+|v_{2}|^{p}+\dotsb +|v_{n}|^{p}{\bigr )}^{1/p},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where the vi are the components of vector v. Except for the case p = 2, the p-norm is not an inner product norm, because it does not satisfy the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Parallelogram_law?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Parallelogram law">parallelogram law</a>. The triangle inequality for general values of p is called <a href="https://en-m-wikipedia-org.translate.goog/wiki/Minkowski%27s_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Minkowski's inequality">Minkowski's inequality</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Saxe-18">[18]</a> It takes the form: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u+v\|_{p}\leq \|u\|_{p}+\|v\|_{p}\ .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> + </mo> <mi> v </mi> <msub> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> p </mi> </mrow> </msub> <mo> ≤ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <msub> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> p </mi> </mrow> </msub> <mo> + </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <msub> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> p </mi> </mrow> </msub> <mtext>   </mtext> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|u+v\|_{p}\leq \|u\|_{p}+\|v\|_{p}\ .} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/588f5e42b6c4b9c7f7143b0e89f791733052bdc0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.074ex; height:3.009ex;" alt="{\displaystyle \|u+v\|_{p}\leq \|u\|_{p}+\|v\|_{p}\ .}"> </noscript><span class="lazy-image-placeholder" style="width: 25.074ex;height: 3.009ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/588f5e42b6c4b9c7f7143b0e89f791733052bdc0" data-alt="{\displaystyle \|u+v\|_{p}\leq \|u\|_{p}+\|v\|_{p}\ .}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <h2 id="Metric_space">Metric space</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=12&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Metric space" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> In a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Metric_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Metric space">metric space</a> M with metric d, the triangle inequality is a requirement upon <a href="https://en-m-wikipedia-org.translate.goog/wiki/Metric_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Definition" class="mw-redirect" title="Metric (mathematics)">distance</a>: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,\ C)\leq d(A,\ B)+d(B,\ C)\ ,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mtext>   </mtext> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo> ≤ </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mtext>   </mtext> <mi> B </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> B </mi> <mo> , </mo> <mtext>   </mtext> <mi> C </mi> <mo stretchy="false"> ) </mo> <mtext>   </mtext> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle d(A,\ C)\leq d(A,\ B)+d(B,\ C)\ ,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce68d5d5bba49599b2ac9186323bbdd4568a6ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.633ex; height:2.843ex;" alt="{\displaystyle d(A,\ C)\leq d(A,\ B)+d(B,\ C)\ ,}"> </noscript><span class="lazy-image-placeholder" style="width: 31.633ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce68d5d5bba49599b2ac9186323bbdd4568a6ec" data-alt="{\displaystyle d(A,\ C)\leq d(A,\ B)+d(B,\ C)\ ,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> for all points A, B, and C in M. That is, the distance from A to C is at most as large as the sum of the distance from A to B and the distance from B to C. The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any <a href="https://en-m-wikipedia-org.translate.goog/wiki/Limit_of_a_sequence?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Limit of a sequence">convergent sequence</a> in a metric space is a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Cauchy_sequence?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cauchy sequence">Cauchy sequence</a> is a direct consequence of the triangle inequality, because if we choose any xn and xm such that d(xn, x) < ε/2 and d(xm, x) < ε/2, where ε > 0 is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, d(xn, xm) ≤ d(xn, x) + d(xm, x) < ε/2 + ε/2 = ε, so that the sequence {xn} is a Cauchy sequence, by definition. This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via d(u, v) ≔ ‖u − v‖, with u − v being the vector pointing from point v to u. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <h2 id="Reverse_triangle_inequality">Reverse triangle inequality</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=13&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Reverse triangle inequality" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> The reverse triangle inequality is an equivalent alternative formulation of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is:<a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-inequality-19">[19]</a> <dl> <dd> Any side of a triangle is greater than or equal to the difference between the other two sides. </dd> </dl> In the case of a normed vector space, the statement is: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\big |}\|u\|-\|v\|{\big |}\leq \|u-v\|,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> | </mo> </mrow> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> − </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> | </mo> </mrow> </mrow> <mo> ≤ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> − </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\big |}\|u\|-\|v\|{\big |}\leq \|u-v\|,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edb25ae558fa1993b75e6c9c7bdee14a8a9e75f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.609ex; height:3.176ex;" alt="{\displaystyle {\big |}\|u\|-\|v\|{\big |}\leq \|u-v\|,}"> </noscript><span class="lazy-image-placeholder" style="width: 22.609ex;height: 3.176ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edb25ae558fa1993b75e6c9c7bdee14a8a9e75f0" data-alt="{\displaystyle {\big |}\|u\|-\|v\|{\big |}\leq \|u-v\|,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> or for metric spaces, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |d(A,C)-d(B,C)|\leq d(A,B)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> B </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> ≤ </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> B </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle |d(A,C)-d(B,C)|\leq d(A,B)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d284ac8bc2efbe211f17ad04a9888265f2919ee7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.957ex; height:2.843ex;" alt="{\displaystyle |d(A,C)-d(B,C)|\leq d(A,B)}"> </noscript><span class="lazy-image-placeholder" style="width: 29.957ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d284ac8bc2efbe211f17ad04a9888265f2919ee7" data-alt="{\displaystyle |d(A,C)-d(B,C)|\leq d(A,B)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . This implies that the norm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\cdot \|}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mo> ⋅ </mo> <mo fence="false" stretchy="false"> ‖ </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|\cdot \|} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113f0d8fe6108fc1c5e9802f7c3f634f5480b3d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.004ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|}"> </noscript><span class="lazy-image-placeholder" style="width: 4.004ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113f0d8fe6108fc1c5e9802f7c3f634f5480b3d1" data-alt="{\displaystyle \|\cdot \|}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  as well as the distance-from- <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"> </noscript><span class="lazy-image-placeholder" style="width: 1.088ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" data-alt="{\displaystyle z}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(z,\cdot )}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> z </mi> <mo> , </mo> <mo> ⋅ </mo> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle d(z,\cdot )} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/934a1c521cfac5b2e36f517b47fc97e63e043d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.794ex; height:2.843ex;" alt="{\displaystyle d(z,\cdot )}"> </noscript><span class="lazy-image-placeholder" style="width: 5.794ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/934a1c521cfac5b2e36f517b47fc97e63e043d25" data-alt="{\displaystyle d(z,\cdot )}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  are <a href="https://en-m-wikipedia-org.translate.goog/wiki/Lipschitz_continuity?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Lipschitz continuity">Lipschitz continuous</a> with Lipschitz constant 1, and therefore are in particular <a href="https://en-m-wikipedia-org.translate.goog/wiki/Uniform_continuity?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Uniform continuity">uniformly continuous</a>. The proof of the reverse triangle inequality from the usual one uses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|v-u\|=\|{-}1(u-v)\|=|{-}1|\cdot \|u-v\|=\|u-v\|}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <mo> − </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> = </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> </mrow> <mn> 1 </mn> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> − </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> </mrow> <mn> 1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> ⋅ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> − </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> = </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> − </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|v-u\|=\|{-}1(u-v)\|=|{-}1|\cdot \|u-v\|=\|u-v\|} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e73a0f04dce3eb41e64267de384f5678f1e13a36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.509ex; height:2.843ex;" alt="{\displaystyle \|v-u\|=\|{-}1(u-v)\|=|{-}1|\cdot \|u-v\|=\|u-v\|}"> </noscript><span class="lazy-image-placeholder" style="width: 50.509ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e73a0f04dce3eb41e64267de384f5678f1e13a36" data-alt="{\displaystyle \|v-u\|=\|{-}1(u-v)\|=|{-}1|\cdot \|u-v\|=\|u-v\|}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  to find: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u\|=\|(u-v)+v\|\leq \|u-v\|+\|v\|\Rightarrow \|u\|-\|v\|\leq \|u-v\|,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> = </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> − </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> ≤ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> − </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> + </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo stretchy="false"> ⇒ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> − </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> ≤ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> − </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|u\|=\|(u-v)+v\|\leq \|u-v\|+\|v\|\Rightarrow \|u\|-\|v\|\leq \|u-v\|,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684c58b4a9d97760668342c50080ebbfba3f7df8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.096ex; height:2.843ex;" alt="{\displaystyle \|u\|=\|(u-v)+v\|\leq \|u-v\|+\|v\|\Rightarrow \|u\|-\|v\|\leq \|u-v\|,}"> </noscript><span class="lazy-image-placeholder" style="width: 62.096ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684c58b4a9d97760668342c50080ebbfba3f7df8" data-alt="{\displaystyle \|u\|=\|(u-v)+v\|\leq \|u-v\|+\|v\|\Rightarrow \|u\|-\|v\|\leq \|u-v\|,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|v\|=\|(v-u)+u\|\leq \|v-u\|+\|u\|\Rightarrow \|u\|-\|v\|\geq -\|u-v\|,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> = </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mo stretchy="false"> ( </mo> <mi> v </mi> <mo> − </mo> <mi> u </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> ≤ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <mo> − </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> + </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo stretchy="false"> ⇒ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> − </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> ≥ </mo> <mo> − </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> − </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|v\|=\|(v-u)+u\|\leq \|v-u\|+\|u\|\Rightarrow \|u\|-\|v\|\geq -\|u-v\|,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cabc473b44ba4bc4ec423f25f6c83f646d3ef346" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.107ex; height:2.843ex;" alt="{\displaystyle \|v\|=\|(v-u)+u\|\leq \|v-u\|+\|u\|\Rightarrow \|u\|-\|v\|\geq -\|u-v\|,}"> </noscript><span class="lazy-image-placeholder" style="width: 64.107ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cabc473b44ba4bc4ec423f25f6c83f646d3ef346" data-alt="{\displaystyle \|v\|=\|(v-u)+u\|\leq \|v-u\|+\|u\|\Rightarrow \|u\|-\|v\|\geq -\|u-v\|,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> Combining these two statements gives: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\|u-v\|\leq \|u\|-\|v\|\leq \|u-v\|\Rightarrow {\big |}\|u\|-\|v\|{\big |}\leq \|u-v\|.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> − </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> − </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> ≤ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> − </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> ≤ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> − </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo stretchy="false"> ⇒ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> | </mo> </mrow> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> − </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> | </mo> </mrow> </mrow> <mo> ≤ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> − </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle -\|u-v\|\leq \|u\|-\|v\|\leq \|u-v\|\Rightarrow {\big |}\|u\|-\|v\|{\big |}\leq \|u-v\|.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc02431fc3c4c6c6bc13bb0859893d4fa3fff7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:59.421ex; height:3.176ex;" alt="{\displaystyle -\|u-v\|\leq \|u\|-\|v\|\leq \|u-v\|\Rightarrow {\big |}\|u\|-\|v\|{\big |}\leq \|u-v\|.}"> </noscript><span class="lazy-image-placeholder" style="width: 59.421ex;height: 3.176ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc02431fc3c4c6c6bc13bb0859893d4fa3fff7b" data-alt="{\displaystyle -\|u-v\|\leq \|u\|-\|v\|\leq \|u-v\|\Rightarrow {\big |}\|u\|-\|v\|{\big |}\leq \|u-v\|.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> In the converse, the proof of the triangle inequality from the reverse triangle inequality works in two cases: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u+v\|-\|u\|\geq 0,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> − </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> ≥ </mo> <mn> 0 </mn> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|u+v\|-\|u\|\geq 0,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1425124b7fa8dbf3739641778fe8b31603bf746" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.025ex; height:2.843ex;" alt="{\displaystyle \|u+v\|-\|u\|\geq 0,}"> </noscript><span class="lazy-image-placeholder" style="width: 19.025ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1425124b7fa8dbf3739641778fe8b31603bf746" data-alt="{\displaystyle \|u+v\|-\|u\|\geq 0,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  then by the reverse triangle inequality, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u+v\|-\|u\|={\big |}\|u+v\|-\|u\|{\big |}\leq \|(u+v)-u\|=\|v\|\Rightarrow \|u+v\|\leq \|u\|+\|v\|}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> − </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> | </mo> </mrow> </mrow> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> − </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em"> | </mo> </mrow> </mrow> <mo> ≤ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> = </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo stretchy="false"> ⇒ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> ≤ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> + </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|u+v\|-\|u\|={\big |}\|u+v\|-\|u\|{\big |}\leq \|(u+v)-u\|=\|v\|\Rightarrow \|u+v\|\leq \|u\|+\|v\|} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee527bcb597218840aaaa44a539bf183dcc8cebb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:80.161ex; height:3.176ex;" alt="{\displaystyle \|u+v\|-\|u\|={\big |}\|u+v\|-\|u\|{\big |}\leq \|(u+v)-u\|=\|v\|\Rightarrow \|u+v\|\leq \|u\|+\|v\|}"> </noscript><span class="lazy-image-placeholder" style="width: 80.161ex;height: 3.176ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee527bcb597218840aaaa44a539bf183dcc8cebb" data-alt="{\displaystyle \|u+v\|-\|u\|={\big |}\|u+v\|-\|u\|{\big |}\leq \|(u+v)-u\|=\|v\|\Rightarrow \|u+v\|\leq \|u\|+\|v\|}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u+v\|-\|u\|<0,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> − </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> < </mo> <mn> 0 </mn> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|u+v\|-\|u\|<0,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4df50b2952621e91d1c362d3f7052cee2bf49619" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.025ex; height:2.843ex;" alt="{\displaystyle \|u+v\|-\|u\|<0,}"> </noscript><span class="lazy-image-placeholder" style="width: 19.025ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4df50b2952621e91d1c362d3f7052cee2bf49619" data-alt="{\displaystyle \|u+v\|-\|u\|<0,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  then trivially <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u\|+\|v\|\geq \|u\|>\|u+v\|}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> + </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> ≥ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> > </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|u\|+\|v\|\geq \|u\|>\|u+v\|} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3541ce84754accbad08c457962d526450b82b62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.422ex; height:2.843ex;" alt="{\displaystyle \|u\|+\|v\|\geq \|u\|>\|u+v\|}"> </noscript><span class="lazy-image-placeholder" style="width: 27.422ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3541ce84754accbad08c457962d526450b82b62" data-alt="{\displaystyle \|u\|+\|v\|\geq \|u\|>\|u+v\|}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  by the nonnegativity of the norm. Thus, in both cases, we find that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u\|+\|v\|\geq \|u+v\|}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> + </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> ≥ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|u\|+\|v\|\geq \|u+v\|} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57ed93fb79539298b5f8e000767a0584c7435a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.669ex; height:2.843ex;" alt="{\displaystyle \|u\|+\|v\|\geq \|u+v\|}"> </noscript><span class="lazy-image-placeholder" style="width: 20.669ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57ed93fb79539298b5f8e000767a0584c7435a0f" data-alt="{\displaystyle \|u\|+\|v\|\geq \|u+v\|}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . For metric spaces, the proof of the reverse triangle inequality is found similarly by: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,B)+d(B,C)\geq d(A,C)\Rightarrow d(A,B)\geq d(A,C)-d(B,C)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> B </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> B </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo> ≥ </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ⇒ </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> B </mi> <mo stretchy="false"> ) </mo> <mo> ≥ </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> B </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle d(A,B)+d(B,C)\geq d(A,C)\Rightarrow d(A,B)\geq d(A,C)-d(B,C)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb55931d9fddf2dae3f46da11ba7c09b0c0fdb4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.94ex; height:2.843ex;" alt="{\displaystyle d(A,B)+d(B,C)\geq d(A,C)\Rightarrow d(A,B)\geq d(A,C)-d(B,C)}"> </noscript><span class="lazy-image-placeholder" style="width: 60.94ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb55931d9fddf2dae3f46da11ba7c09b0c0fdb4f" data-alt="{\displaystyle d(A,B)+d(B,C)\geq d(A,C)\Rightarrow d(A,B)\geq d(A,C)-d(B,C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(C,A)+d(A,B)\geq d(C,B)\Rightarrow d(A,B)\geq d(B,C)-d(A,C)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> C </mi> <mo> , </mo> <mi> A </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> B </mi> <mo stretchy="false"> ) </mo> <mo> ≥ </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> C </mi> <mo> , </mo> <mi> B </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ⇒ </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> B </mi> <mo stretchy="false"> ) </mo> <mo> ≥ </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> B </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle d(C,A)+d(A,B)\geq d(C,B)\Rightarrow d(A,B)\geq d(B,C)-d(A,C)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/356a53ddd5dac74f1e86e34bbb56c183b5e33de3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.94ex; height:2.843ex;" alt="{\displaystyle d(C,A)+d(A,B)\geq d(C,B)\Rightarrow d(A,B)\geq d(B,C)-d(A,C)}"> </noscript><span class="lazy-image-placeholder" style="width: 60.94ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/356a53ddd5dac74f1e86e34bbb56c183b5e33de3" data-alt="{\displaystyle d(C,A)+d(A,B)\geq d(C,B)\Rightarrow d(A,B)\geq d(B,C)-d(A,C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Putting these equations together we find: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,B)\geq |d(A,C)-d(B,C)|}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> B </mi> <mo stretchy="false"> ) </mo> <mo> ≥ </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> B </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle d(A,B)\geq |d(A,C)-d(B,C)|} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d13e3b0739948fe8c1785bc86f57f15d4974aa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.957ex; height:2.843ex;" alt="{\displaystyle d(A,B)\geq |d(A,C)-d(B,C)|}"> </noscript><span class="lazy-image-placeholder" style="width: 29.957ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d13e3b0739948fe8c1785bc86f57f15d4974aa5" data-alt="{\displaystyle d(A,B)\geq |d(A,C)-d(B,C)|}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  And in the converse, beginning from the reverse triangle inequality, we can again use two cases: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,C)-d(B,C)\geq 0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> B </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo> ≥ </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle d(A,C)-d(B,C)\geq 0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86143c8d7072555ba7bbc59b399f17ef5eec5d00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.259ex; height:2.843ex;" alt="{\displaystyle d(A,C)-d(B,C)\geq 0}"> </noscript><span class="lazy-image-placeholder" style="width: 22.259ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86143c8d7072555ba7bbc59b399f17ef5eec5d00" data-alt="{\displaystyle d(A,C)-d(B,C)\geq 0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,B)\geq |d(A,C)-d(B,C)|=d(A,C)-d(B,C)\Rightarrow d(A,B)+d(B,C)\geq d(A,C)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> B </mi> <mo stretchy="false"> ) </mo> <mo> ≥ </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> B </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> = </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> B </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ⇒ </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> B </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> B </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo> ≥ </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle d(A,B)\geq |d(A,C)-d(B,C)|=d(A,C)-d(B,C)\Rightarrow d(A,B)+d(B,C)\geq d(A,C)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42063e2fac26954702d436abaabd543887c776ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:83.331ex; height:2.843ex;" alt="{\displaystyle d(A,B)\geq |d(A,C)-d(B,C)|=d(A,C)-d(B,C)\Rightarrow d(A,B)+d(B,C)\geq d(A,C)}"> </noscript><span class="lazy-image-placeholder" style="width: 83.331ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42063e2fac26954702d436abaabd543887c776ce" data-alt="{\displaystyle d(A,B)\geq |d(A,C)-d(B,C)|=d(A,C)-d(B,C)\Rightarrow d(A,B)+d(B,C)\geq d(A,C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,C)-d(B,C)<0,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> B </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo> < </mo> <mn> 0 </mn> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle d(A,C)-d(B,C)<0,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7212db02fffe811c8c65708c1f7e549d360cba11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.906ex; height:2.843ex;" alt="{\displaystyle d(A,C)-d(B,C)<0,}"> </noscript><span class="lazy-image-placeholder" style="width: 22.906ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7212db02fffe811c8c65708c1f7e549d360cba11" data-alt="{\displaystyle d(A,C)-d(B,C)<0,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,B)+d(B,C)\geq d(B,C)>d(A,C)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> B </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> B </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo> ≥ </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> B </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo> > </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle d(A,B)+d(B,C)\geq d(B,C)>d(A,C)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a56962bfb75385d74ee3823a0fc0b97312e830c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.351ex; height:2.843ex;" alt="{\displaystyle d(A,B)+d(B,C)\geq d(B,C)>d(A,C)}"> </noscript><span class="lazy-image-placeholder" style="width: 39.351ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a56962bfb75385d74ee3823a0fc0b97312e830c" data-alt="{\displaystyle d(A,B)+d(B,C)\geq d(B,C)>d(A,C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  again by the nonnegativity of the metric. Thus, in both cases, we find that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,B)+d(B,C)\geq d(A,C)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> B </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> B </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> <mo> ≥ </mo> <mi> d </mi> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo> , </mo> <mi> C </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle d(A,B)+d(B,C)\geq d(A,C)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85366f1965268ae63a5444600dcbdfb3155b80b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.663ex; height:2.843ex;" alt="{\displaystyle d(A,B)+d(B,C)\geq d(A,C)}"> </noscript><span class="lazy-image-placeholder" style="width: 28.663ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85366f1965268ae63a5444600dcbdfb3155b80b" data-alt="{\displaystyle d(A,B)+d(B,C)\geq d(A,C)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <h2 id="Triangle_inequality_for_cosine_similarity">Triangle inequality for cosine similarity</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=14&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Triangle inequality for cosine similarity" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"> <div role="note" class="hatnote navigation-not-searchable"> Further information: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Cosine_similarity?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cosine similarity">Cosine similarity</a> </div> By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that<a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-20">[20]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sim} (u,w)\geq \operatorname {sim} (u,v)\cdot \operatorname {sim} (v,w)-{\sqrt {\left(1-\operatorname {sim} (u,v)^{2}\right)\cdot \left(1-\operatorname {sim} (v,w)^{2}\right)}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> sim </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> w </mi> <mo stretchy="false"> ) </mo> <mo> ≥ </mo> <mi> sim </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mi> sim </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> v </mi> <mo> , </mo> <mi> w </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> − </mo> <mi> sim </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> − </mo> <mi> sim </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> v </mi> <mo> , </mo> <mi> w </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \operatorname {sim} (u,w)\geq \operatorname {sim} (u,v)\cdot \operatorname {sim} (v,w)-{\sqrt {\left(1-\operatorname {sim} (u,v)^{2}\right)\cdot \left(1-\operatorname {sim} (v,w)^{2}\right)}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70765841cee7b5a0ffca1a1fc60140583ea0852f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:71.198ex; height:4.843ex;" alt="{\displaystyle \operatorname {sim} (u,w)\geq \operatorname {sim} (u,v)\cdot \operatorname {sim} (v,w)-{\sqrt {\left(1-\operatorname {sim} (u,v)^{2}\right)\cdot \left(1-\operatorname {sim} (v,w)^{2}\right)}}}"> </noscript><span class="lazy-image-placeholder" style="width: 71.198ex;height: 4.843ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70765841cee7b5a0ffca1a1fc60140583ea0852f" data-alt="{\displaystyle \operatorname {sim} (u,w)\geq \operatorname {sim} (u,v)\cdot \operatorname {sim} (v,w)-{\sqrt {\left(1-\operatorname {sim} (u,v)^{2}\right)\cdot \left(1-\operatorname {sim} (v,w)^{2}\right)}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sim} (u,w)\leq \operatorname {sim} (u,v)\cdot \operatorname {sim} (v,w)+{\sqrt {\left(1-\operatorname {sim} (u,v)^{2}\right)\cdot \left(1-\operatorname {sim} (v,w)^{2}\right)}}\,.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> sim </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> w </mi> <mo stretchy="false"> ) </mo> <mo> ≤ </mo> <mi> sim </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> <mo> ⋅ </mo> <mi> sim </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> v </mi> <mo> , </mo> <mi> w </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> − </mo> <mi> sim </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⋅ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> − </mo> <mi> sim </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> v </mi> <mo> , </mo> <mi> w </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </msqrt> </mrow> <mspace width="thinmathspace"></mspace> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \operatorname {sim} (u,w)\leq \operatorname {sim} (u,v)\cdot \operatorname {sim} (v,w)+{\sqrt {\left(1-\operatorname {sim} (u,v)^{2}\right)\cdot \left(1-\operatorname {sim} (v,w)^{2}\right)}}\,.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8290849016fbe65ebca0530eb04055c907ed4705" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:72.232ex; height:4.843ex;" alt="{\displaystyle \operatorname {sim} (u,w)\leq \operatorname {sim} (u,v)\cdot \operatorname {sim} (v,w)+{\sqrt {\left(1-\operatorname {sim} (u,v)^{2}\right)\cdot \left(1-\operatorname {sim} (v,w)^{2}\right)}}\,.}"> </noscript><span class="lazy-image-placeholder" style="width: 72.232ex;height: 4.843ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8290849016fbe65ebca0530eb04055c907ed4705" data-alt="{\displaystyle \operatorname {sim} (u,w)\leq \operatorname {sim} (u,v)\cdot \operatorname {sim} (v,w)+{\sqrt {\left(1-\operatorname {sim} (u,v)^{2}\right)\cdot \left(1-\operatorname {sim} (v,w)^{2}\right)}}\,.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  With these formulas, one needs to compute a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Square_root?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Square root">square root</a> for each triple of vectors {u, v, w} that is examined rather than arccos(sim(u,v)) for each pair of vectors {u, v} examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <h2 id="Reversal_in_Minkowski_space">Reversal in Minkowski space</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=15&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Reversal in Minkowski space" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> The <a href="https://en-m-wikipedia-org.translate.goog/wiki/Minkowski_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Minkowski space">Minkowski space</a> metric <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\mu \nu }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> η </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> μ </mi> <mi> ν </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \eta _{\mu \nu }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a8fb25ea6ee5d591c2d819519401a871ee04bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.25ex; height:2.343ex;" alt="{\displaystyle \eta _{\mu \nu }}"> </noscript><span class="lazy-image-placeholder" style="width: 3.25ex;height: 2.343ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a8fb25ea6ee5d591c2d819519401a871ee04bc6" data-alt="{\displaystyle \eta _{\mu \nu }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is not positive-definite, which means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u\|^{2}=\eta _{\mu \nu }u^{\mu }u^{\nu }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <msup> <mo fence="false" stretchy="false"> ‖ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> = </mo> <msub> <mi> η </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> μ </mi> <mi> ν </mi> </mrow> </msub> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> μ </mi> </mrow> </msup> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> ν </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|u\|^{2}=\eta _{\mu \nu }u^{\mu }u^{\nu }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d17894e28281ee55d2d04351fba5f9e78164344d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.044ex; height:3.343ex;" alt="{\displaystyle \|u\|^{2}=\eta _{\mu \nu }u^{\mu }u^{\nu }}"> </noscript><span class="lazy-image-placeholder" style="width: 16.044ex;height: 3.343ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d17894e28281ee55d2d04351fba5f9e78164344d" data-alt="{\displaystyle \|u\|^{2}=\eta _{\mu \nu }u^{\mu }u^{\nu }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  can have either sign or vanish, even if the vector u is non-zero. Moreover, if u and v are both timelike vectors lying in the future light cone, the triangle inequality is reversed: <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|u+v\|\geq \|u\|+\|v\|.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> ≥ </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> u </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> + </mo> <mo fence="false" stretchy="false"> ‖ </mo> <mi> v </mi> <mo fence="false" stretchy="false"> ‖ </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \|u+v\|\geq \|u\|+\|v\|.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b13717002f66e070c812be642d0feeaa777659ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.315ex; height:2.843ex;" alt="{\displaystyle \|u+v\|\geq \|u\|+\|v\|.}"> </noscript><span class="lazy-image-placeholder" style="width: 21.315ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b13717002f66e070c812be642d0feeaa777659ac" data-alt="{\displaystyle \|u+v\|\geq \|u\|+\|v\|.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> A physical example of this inequality is the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Twin_paradox?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Twin paradox">twin paradox</a> in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special_relativity?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special relativity">special relativity</a>. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n+1} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"> </noscript><span class="lazy-image-placeholder" style="width: 5.398ex;height: 2.343ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" data-alt="{\displaystyle n+1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  dimensions for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> n </mi> <mo> ≥ </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle n\geq 1} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"> </noscript><span class="lazy-image-placeholder" style="width: 5.656ex;height: 2.343ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" data-alt="{\displaystyle n\geq 1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . If the plane defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> u </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" data-alt="{\displaystyle u}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> </noscript><span class="lazy-image-placeholder" style="width: 1.128ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" data-alt="{\displaystyle v}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is space-like (and therefore a Euclidean subspace) then the usual triangle inequality holds. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"> <h2 id="See_also">See also</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=16&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-7 collapsible-block" id="mf-section-7"> <ul> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Subadditivity?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Subadditivity">Subadditivity</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Minkowski_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Minkowski inequality">Minkowski inequality</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Ptolemy%27s_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ptolemy's inequality">Ptolemy's inequality</a></li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"> <h2 id="Notes">Notes</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=17&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Notes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-8 collapsible-block" id="mf-section-8"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style> <div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-1">^</a> Wolfram MathWorld – <a rel="nofollow" class="external free" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://mathworld.wolfram.com/TriangleInequality.html">http://mathworld.wolfram.com/TriangleInequality.html</a></li> <li id="cite_note-Khamsi-2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Khamsi_2-0">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMohamed_A._KhamsiWilliam_A._Kirk2001" class="citation book cs1">Mohamed A. Khamsi; William A. Kirk (2001). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3D4qXbEpAK5eUC%26pg%3DPA8">"§1.4 The triangle inequality in Rn"</a>. An introduction to metric spaces and fixed point theory. Wiley-IEEE. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-471-41825-0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-471-41825-0"><bdi>0-471-41825-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A71.4+The+triangle+inequality+in+%3Cspan+class%3D%22texhtml+%22+%3ER%3Csup%3En%3C%2Fsup%3E%3C%2Fspan%3E&rft.btitle=An+introduction+to+metric+spaces+and+fixed+point+theory&rft.pub=Wiley-IEEE&rft.date=2001&rft.isbn=0-471-41825-0&rft.au=Mohamed+A.+Khamsi&rft.au=William+A.+Kirk&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4qXbEpAK5eUC%26pg%3DPA8&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li id="cite_note-3"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-3">^</a> for instance, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacobs1974" class="citation cs2">Jacobs, Harold R. (1974), Geometry, W. H. Freeman & Co., p. 246, <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-7167-0456-0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-7167-0456-0"><bdi>0-7167-0456-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry&rft.pages=246&rft.pub=W.+H.+Freeman+%26+Co.&rft.date=1974&rft.isbn=0-7167-0456-0&rft.aulast=Jacobs&rft.aufirst=Harold+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li id="cite_note-Ramos-4"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Ramos_4-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOliver_BrockJeff_TrinkleFabio_Ramos2009" class="citation book cs1">Oliver Brock; Jeff Trinkle; Fabio Ramos (2009). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DfvCaQfBQ7qEC%26pg%3DPA195">Robotics: Science and Systems IV</a>. MIT Press. p. 195. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-262-51309-8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-262-51309-8"><bdi>978-0-262-51309-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Robotics%3A+Science+and+Systems+IV&rft.pages=195&rft.pub=MIT+Press&rft.date=2009&rft.isbn=978-0-262-51309-8&rft.au=Oliver+Brock&rft.au=Jeff+Trinkle&rft.au=Fabio+Ramos&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfvCaQfBQ7qEC%26pg%3DPA195&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li id="cite_note-Ramsay-5"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Ramsay_5-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArlan_RamsayRobert_D._Richtmyer1995" class="citation book cs1">Arlan Ramsay; Robert D. Richtmyer (1995). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/introductiontohy0000rams">Introduction to hyperbolic geometry</a>. Springer. p. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/introductiontohy0000rams/page/17">17</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-387-94339-0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-387-94339-0"><bdi>0-387-94339-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+hyperbolic+geometry&rft.pages=17&rft.pub=Springer&rft.date=1995&rft.isbn=0-387-94339-0&rft.au=Arlan+Ramsay&rft.au=Robert+D.+Richtmyer&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontohy0000rams&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li id="cite_note-Jacobs-6"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Jacobs_6-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarold_R._Jacobs2003" class="citation book cs1">Harold R. Jacobs (2003). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DXhQRgZRDDq0C%26pg%3DPA201">Geometry: seeing, doing, understanding</a> (3rd ed.). Macmillan. p. 201. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-7167-4361-2?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-7167-4361-2"><bdi>0-7167-4361-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry%3A+seeing%2C+doing%2C+understanding&rft.pages=201&rft.edition=3rd&rft.pub=Macmillan&rft.date=2003&rft.isbn=0-7167-4361-2&rft.au=Harold+R.+Jacobs&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXhQRgZRDDq0C%26pg%3DPA201&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li id="cite_note-Joyce-7"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Joyce_7-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_E._Joyce1997" class="citation web cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/David_E._Joyce_(mathematician)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="David E. Joyce (mathematician)">David E. Joyce</a> (1997). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI20.html">"Euclid's elements, Book 1, Proposition 20"</a>. Euclid's elements. Dept. Math and Computer Science, Clark University. Retrieved 2010-06-25.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Euclid%27s+elements&rft.atitle=Euclid%27s+elements%2C+Book+1%2C+Proposition+20&rft.date=1997&rft.au=David+E.+Joyce&rft_id=http%3A%2F%2Faleph0.clarku.edu%2F~djoyce%2Fjava%2Felements%2FbookI%2FpropI20.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li id="cite_note-8"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-8">^</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/American_Mathematical_Monthly?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>, pp. 49-50, 1954.</li> <li id="cite_note-Palmer-9"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Palmer_9-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClaude_Irwin_Palmer1919" class="citation book cs1">Claude Irwin Palmer (1919). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/practicalmathema00palmiala">Practical mathematics for home study: being the essentials of arithmetic, geometry, algebra and trigonometry</a>. McGraw-Hill. p. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/practicalmathema00palmiala/page/422">422</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Practical+mathematics+for+home+study%3A+being+the+essentials+of+arithmetic%2C+geometry%2C+algebra+and+trigonometry&rft.pages=422&rft.pub=McGraw-Hill&rft.date=1919&rft.au=Claude+Irwin+Palmer&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fpracticalmathema00palmiala&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li id="cite_note-Zawaira-10"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Zawaira_10-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlexander_ZawairaGavin_Hitchcock2009" class="citation book cs1">Alexander Zawaira; Gavin Hitchcock (2009). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DA21T73sqZ3AC%26pg%3DPA30">"Lemma 1: In a right-angled triangle the hypotenuse is greater than either of the other two sides"</a>. A primer for mathematics competitions. Oxford University Press. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-19-953988-8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-19-953988-8"><bdi>978-0-19-953988-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Lemma+1%3A+In+a+right-angled+triangle+the+hypotenuse+is+greater+than+either+of+the+other+two+sides&rft.btitle=A+primer+for+mathematics+competitions&rft.pub=Oxford+University+Press&rft.date=2009&rft.isbn=978-0-19-953988-8&rft.au=Alexander+Zawaira&rft.au=Gavin+Hitchcock&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DA21T73sqZ3AC%26pg%3DPA30&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li id="cite_note-11"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolfram|Alpha" class="citation journal cs1">Wolfram|Alpha. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.wolframalpha.com/input/?i%3Dsolve%25200%253Ca%253C2a%252B3d%252C%25200%253Ca%252Bd%253C2a%252B2d%252C%25200%253Ca%252B2d%253C2a%252Bd%26t%3Dff3tb01">"input: solve 0<a<2a+3d, 0<a+d<2a+2d, 0<a+2d<2a+d,"</a>. Wolfram Research. Retrieved 2010-09-07.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Wolfram+Research&rft.atitle=input%3A+solve+0%3Ca%3C2a%2B3d%2C+0%3Ca%2Bd%3C2a%2B2d%2C+0%3Ca%2B2d%3C2a%2Bd%2C&rft.au=Wolfram%7CAlpha&rft_id=http%3A%2F%2Fwww.wolframalpha.com%2Finput%2F%3Fi%3Dsolve%25200%253Ca%253C2a%252B3d%252C%25200%253Ca%252Bd%253C2a%252B2d%252C%25200%253Ca%252B2d%253C2a%252Bd%26t%3Dff3tb01&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li id="cite_note-12"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-12">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolfram|Alpha" class="citation journal cs1">Wolfram|Alpha. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://wolframalpha.com/input?i%3Dsolve%2B0%253Ca%253Car%252Bar%255E2%252C%2B0%253Car%253Ca%252Bar%255E2%252C%2B0%253Car%255E2%253Ca%252Bar">"input: solve 0<a<ar+ar2, 0<ar<a+ar2, 0<ar2<a+ar"</a>. Wolfram Research. Retrieved 2010-09-07.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Wolfram+Research&rft.atitle=input%3A+solve+0%3Ca%3Car%2Bar%3Csup%3E2%3C%2Fsup%3E%2C+0%3Car%3Ca%2Bar%3Csup%3E2%3C%2Fsup%3E%2C+0%3Car%3Csup%3E2%3C%2Fsup%3E%3Ca%2Bar&rft.au=Wolfram%7CAlpha&rft_id=http%3A%2F%2Fwolframalpha.com%2Finput%3Fi%3Dsolve%2B0%253Ca%253Car%252Bar%5E2%252C%2B0%253Car%253Ca%252Bar%5E2%252C%2B0%253Car%5E2%253Ca%252Bar&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li id="cite_note-13"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolfram|Alpha" class="citation journal cs1">Wolfram|Alpha. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.wolframalpha.com/input/?i%3Dsolve%2520%257B0%253Ca%253Ca*r%252Ba*r%255E2%252Ba*r%255E3%252C%25200%253Ca*r%255E3%253Ca%252Ba*r%252Ba*r%255E2%257D%26t%3Dff3tb01">"input: solve 0<a<ar+ar2+ar3, 0<ar3<a+ar+ar2"</a>. Wolfram Research. Retrieved 2012-07-29.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Wolfram+Research&rft.atitle=input%3A+solve+0%3Ca%3Car%2Bar%3Csup%3E2%3C%2Fsup%3E%2Bar%3Csup%3E3%3C%2Fsup%3E%2C+0%3Car%3Csup%3E3%3C%2Fsup%3E%3Ca%2Bar%2Bar%3Csup%3E2%3C%2Fsup%3E&rft.au=Wolfram%7CAlpha&rft_id=http%3A%2F%2Fwww.wolframalpha.com%2Finput%2F%3Fi%3Dsolve%2520%7B0%253Ca%253Ca%2Ar%252Ba%2Ar%5E2%252Ba%2Ar%5E3%252C%25200%253Ca%2Ar%5E3%253Ca%252Ba%2Ar%252Ba%2Ar%5E2%7D%26t%3Dff3tb01&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li id="cite_note-14"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-14">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_Stillwell1997" class="citation book cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/John_Stillwell?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="John Stillwell">John Stillwell</a> (1997). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3D4elkHwVS0eUC%26pg%3DPA95">Numbers and Geometry</a>. Springer. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-387-98289-2?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-387-98289-2"><bdi>978-0-387-98289-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numbers+and+Geometry&rft.pub=Springer&rft.date=1997&rft.isbn=978-0-387-98289-2&rft.au=John+Stillwell&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4elkHwVS0eUC%26pg%3DPA95&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"> p. 95.</li> <li id="cite_note-Kress-15"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Kress_15-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRainer_Kress1988" class="citation book cs1">Rainer Kress (1988). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3De7ZmHRIxum0C%26pg%3DPA26">"§3.1: Normed spaces"</a>. Numerical analysis. Springer. p. 26. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-387-98408-9?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-387-98408-9"><bdi>0-387-98408-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A73.1%3A+Normed+spaces&rft.btitle=Numerical+analysis&rft.pages=26&rft.pub=Springer&rft.date=1988&rft.isbn=0-387-98408-9&rft.au=Rainer+Kress&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3De7ZmHRIxum0C%26pg%3DPA26&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li id="cite_note-Stewart-16"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Stewart_16-0">^</a> A proof not requiring separate cases is as follows: Any number is always less than or equal to its own absolute value, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -|u|\leq u\leq |u|}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> ≤ </mo> <mi> u </mi> <mo> ≤ </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle -|u|\leq u\leq |u|} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc6bc7bc5b59e0aee53f393667d6399033d1dc96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.581ex; height:2.843ex;" alt="{\displaystyle -|u|\leq u\leq |u|}"> </noscript><span class="lazy-image-placeholder" style="width: 14.581ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc6bc7bc5b59e0aee53f393667d6399033d1dc96" data-alt="{\displaystyle -|u|\leq u\leq |u|}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -|v|\leq v\leq |v|.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> ≤ </mo> <mi> v </mi> <mo> ≤ </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle -|v|\leq v\leq |v|.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87144c9537a86b47c2e5036a8dae15ee8916f266" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.622ex; height:2.843ex;" alt="{\displaystyle -|v|\leq v\leq |v|.}"> </noscript><span class="lazy-image-placeholder" style="width: 14.622ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87144c9537a86b47c2e5036a8dae15ee8916f266" data-alt="{\displaystyle -|v|\leq v\leq |v|.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Adding these inequalities together, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\bigl (}|u|+|v|{\bigr )}\leq u+v\leq |u|+|v|,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> <mo> ≤ </mo> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mo> ≤ </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle -{\bigl (}|u|+|v|{\bigr )}\leq u+v\leq |u|+|v|,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ebd02905eac0e3481973cc56fa5ace35c647bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.85ex; height:3.176ex;" alt="{\displaystyle -{\bigl (}|u|+|v|{\bigr )}\leq u+v\leq |u|+|v|,}"> </noscript><span class="lazy-image-placeholder" style="width: 31.85ex;height: 3.176ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ebd02905eac0e3481973cc56fa5ace35c647bf" data-alt="{\displaystyle -{\bigl (}|u|+|v|{\bigr )}\leq u+v\leq |u|+|v|,}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  the sum has the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -a\leq b\leq a}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo> − </mo> <mi> a </mi> <mo> ≤ </mo> <mi> b </mi> <mo> ≤ </mo> <mi> a </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle -a\leq b\leq a} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2daabcaed944a0533186ec4a2bb5e6e888162d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.462ex; height:2.343ex;" alt="{\displaystyle -a\leq b\leq a}"> </noscript><span class="lazy-image-placeholder" style="width: 11.462ex;height: 2.343ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2daabcaed944a0533186ec4a2bb5e6e888162d82" data-alt="{\displaystyle -a\leq b\leq a}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=|u|+|v|}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> a </mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a=|u|+|v|} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c202bc46c6d3135a6d5b27812f110f39e79358e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.213ex; height:2.843ex;" alt="{\displaystyle a=|u|+|v|}"> </noscript><span class="lazy-image-placeholder" style="width: 12.213ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c202bc46c6d3135a6d5b27812f110f39e79358e" data-alt="{\displaystyle a=|u|+|v|}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=u+v.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> b </mi> <mo> = </mo> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle b=u+v.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcbe4ad3c281db177a67365c8c5e646c6186a9bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.041ex; height:2.343ex;" alt="{\displaystyle b=u+v.}"> </noscript><span class="lazy-image-placeholder" style="width: 10.041ex;height: 2.343ex;vertical-align: -0.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcbe4ad3c281db177a67365c8c5e646c6186a9bc" data-alt="{\displaystyle b=u+v.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  But this always implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |b|\leq a}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> b </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> ≤ </mo> <mi> a </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle |b|\leq a} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d6de1867d123b96a6bb14f49255d5b7ca6055f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.62ex; height:2.843ex;" alt="{\displaystyle |b|\leq a}"> </noscript><span class="lazy-image-placeholder" style="width: 6.62ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d6de1867d123b96a6bb14f49255d5b7ca6055f2" data-alt="{\displaystyle |b|\leq a}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , or, expanded, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |u+v|\leq |u|+|v|.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mo> + </mo> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> ≤ </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle |u+v|\leq |u|+|v|.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e7dd5741100aa30dcc531841647c8bc5b03e835" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.222ex; height:2.843ex;" alt="{\displaystyle |u+v|\leq |u|+|v|.}"> </noscript><span class="lazy-image-placeholder" style="width: 18.222ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e7dd5741100aa30dcc531841647c8bc5b03e835" data-alt="{\displaystyle |u+v|\leq |u|+|v|.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <div class="paragraphbreak" style="margin-top:0.5em"></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJames_Stewart2008" class="citation book cs1">James Stewart (2008). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/studentsolutions0000stew">Essential Calculus</a>. Thomson Brooks/Cole. p. A10. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-495-10860-3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-495-10860-3"><bdi>978-0-495-10860-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essential+Calculus&rft.pages=A10&rft.pub=Thomson+Brooks%2FCole&rft.date=2008&rft.isbn=978-0-495-10860-3&rft.au=James+Stewart&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fstudentsolutions0000stew&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li id="cite_note-Stillwell-17"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Stillwell_17-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_Stillwell2005" class="citation book cs1">John Stillwell (2005). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/fourpillarsofgeo0000stil">The four pillars of geometry</a>. Springer. p. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/fourpillarsofgeo0000stil/page/80">80</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-387-25530-3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-387-25530-3"><bdi>0-387-25530-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+four+pillars+of+geometry&rft.pages=80&rft.pub=Springer&rft.date=2005&rft.isbn=0-387-25530-3&rft.au=John+Stillwell&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffourpillarsofgeo0000stil&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li id="cite_note-Saxe-18"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Saxe_18-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaren_Saxe2002" class="citation book cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Karen_Saxe?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Karen Saxe">Karen Saxe</a> (2002). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3D0LeWJ74j8GQC%26pg%3DPA61">Beginning functional analysis</a>. Springer. p. 61. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-387-95224-1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-387-95224-1"><bdi>0-387-95224-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Beginning+functional+analysis&rft.pages=61&rft.pub=Springer&rft.date=2002&rft.isbn=0-387-95224-1&rft.au=Karen+Saxe&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0LeWJ74j8GQC%26pg%3DPA61&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li id="cite_note-inequality-19"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-inequality_19-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnonymous1854" class="citation book cs1">Anonymous (1854). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DlTACAAAAQAAJ%26pg%3DPA196">"Exercise I. to proposition XIX"</a>. The popular educator; fourth volume. Ludgate Hill, London: John Cassell. p. 196.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Exercise+I.+to+proposition+XIX&rft.btitle=The+popular+educator%3B+fourth+volume&rft.place=Ludgate+Hill%2C+London&rft.pages=196&rft.pub=John+Cassell&rft.date=1854&rft.au=Anonymous&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlTACAAAAQAAJ%26pg%3DPA196&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li id="cite_note-20"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangle_inequality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchubert2021" class="citation conference cs1">Schubert, Erich (2021). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://arxiv.org/pdf/2107.04071">A Triangle Inequality for Cosine Similarity</a>. International Conference on Similarity Search and Applications. Dortmund: Springer. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007%252F978-3-030-89657-7_3">10.1007/978-3-030-89657-7_3</a>. Retrieved 29 January 2025.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=A+Triangle+Inequality+for+Cosine+Similarity&rft.place=Dortmund&rft.pub=Springer&rft.date=2021&rft_id=info%3Adoi%2F10.1007%2F978-3-030-89657-7_3&rft.aulast=Schubert&rft.aufirst=Erich&rft_id=https%3A%2F%2Farxiv.org%2Fpdf%2F2107.04071&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> </ol> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"> <h2 id="References">References</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Triangle_inequality&action=edit&section=18&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-9 collapsible-block" id="mf-section-9"> <ul> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPedoe1988" class="citation book cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Daniel_Pedoe?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Daniel Pedoe">Pedoe, Daniel</a> (1988). Geometry: A comprehensive course. Dover. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-486-65812-0?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-486-65812-0"><bdi>0-486-65812-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry%3A+A+comprehensive+course&rft.pub=Dover&rft.date=1988&rft.isbn=0-486-65812-0&rft.aulast=Pedoe&rft.aufirst=Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudin1976" class="citation book cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Walter_Rudin?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Walter Rudin">Rudin, Walter</a> (1976). Principles of Mathematical Analysis. New York: <a href="https://en-m-wikipedia-org.translate.goog/wiki/McGraw-Hill?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-07-054235-X?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-07-054235-X"><bdi>0-07-054235-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Mathematical+Analysis&rft.place=New+York&rft.pub=McGraw-Hill&rft.date=1976&rft.isbn=0-07-054235-X&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATriangle+inequality" class="Z3988"></li> </ul>   </section> </div> <noscript> <img 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id="mw-data-after-content"> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://als.wikipedia.org/wiki/Dreiecksungleichung" title="Dreiecksungleichung – Alemannic" lang="gsw" hreflang="gsw" data-title="Dreiecksungleichung" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ar.wikipedia.org/wiki/%25D9%2585%25D8%25AA%25D8%25A8%25D8%25A7%25D9%258A%25D9%2586%25D8%25A9_%25D8%25A7%25D9%2584%25D9%2585%25D8%25AB%25D9%2584%25D8%25AB" title="متباينة المثلث – Arabic" lang="ar" hreflang="ar" data-title="متباينة المثلث" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li> <li class="interlanguage-link interwiki-be mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://be.wikipedia.org/wiki/%25D0%259D%25D1%258F%25D1%2580%25D0%25BE%25D1%259E%25D0%25BD%25D0%25B0%25D1%2581%25D1%2586%25D1%258C_%25D1%2582%25D1%2580%25D0%25BE%25D1%2585%25D0%25B2%25D1%2583%25D0%25B3%25D0%25BE%25D0%25BB%25D1%258C%25D0%25BD%25D1%2596%25D0%25BA%25D0%25B0" title="Няроўнасць трохвугольніка – Belarusian" lang="be" hreflang="be" data-title="Няроўнасць трохвугольніка" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bg.wikipedia.org/wiki/%25D0%259D%25D0%25B5%25D1%2580%25D0%25B0%25D0%25B2%25D0%25B5%25D0%25BD%25D1%2581%25D1%2582%25D0%25B2%25D0%25BE_%25D0%25BD%25D0%25B0_%25D1%2582%25D1%2580%25D0%25B8%25D1%258A%25D0%25B3%25D1%258A%25D0%25BB%25D0%25BD%25D0%25B8%25D0%25BA%25D0%25B0" title="Неравенство на триъгълника – Bulgarian" lang="bg" hreflang="bg" data-title="Неравенство на триъгълника" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ca.wikipedia.org/wiki/Desigualtat_triangular" title="Desigualtat triangular – Catalan" lang="ca" hreflang="ca" data-title="Desigualtat triangular" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li> <li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cv.wikipedia.org/wiki/%25D0%2592%25D0%25B8%25C3%25A7%25D0%25BA%25C4%2595%25D1%2582%25D0%25B5%25D1%2581%25D0%25BB%25C4%2595%25D1%2585_%25D1%2582%25D0%25B0%25D0%25BD%25D0%25BC%25D0%25B0%25D1%2580%25D0%25BB%25C4%2583%25D1%2585%25C4%2595" title="Виçкĕтеслĕх танмарлăхĕ – Chuvash" lang="cv" hreflang="cv" data-title="Виçкĕтеслĕх танмарлăхĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wikipedia.org/wiki/Troj%25C3%25BAheln%25C3%25ADkov%25C3%25A1_nerovnost" title="Trojúhelníková nerovnost – Czech" lang="cs" hreflang="cs" data-title="Trojúhelníková nerovnost" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li> <li class="interlanguage-link interwiki-da mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://da.wikipedia.org/wiki/Trekantsuligheden" title="Trekantsuligheden – Danish" lang="da" hreflang="da" data-title="Trekantsuligheden" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://de.wikipedia.org/wiki/Dreiecksungleichung" title="Dreiecksungleichung – German" lang="de" hreflang="de" data-title="Dreiecksungleichung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li> <li class="interlanguage-link interwiki-et mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://et.wikipedia.org/wiki/Kolmnurga_v%25C3%25B5rratus" title="Kolmnurga võrratus – Estonian" lang="et" hreflang="et" data-title="Kolmnurga võrratus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li> <li class="interlanguage-link interwiki-el mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://el.wikipedia.org/wiki/%25CE%25A4%25CF%2581%25CE%25B9%25CE%25B3%25CF%2589%25CE%25BD%25CE%25B9%25CE%25BA%25CE%25AE_%25CE%25B1%25CE%25BD%25CE%25B9%25CF%2583%25CF%258C%25CF%2584%25CE%25B7%25CF%2584%25CE%25B1" title="Τριγωνική ανισότητα – Greek" lang="el" hreflang="el" data-title="Τριγωνική ανισότητα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://es.wikipedia.org/wiki/Desigualdad_triangular" title="Desigualdad triangular – Spanish" lang="es" hreflang="es" data-title="Desigualdad triangular" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li> <li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eo.wikipedia.org/wiki/Neegala%25C4%25B5o_de_triangulo" title="Neegalaĵo de triangulo – Esperanto" lang="eo" hreflang="eo" data-title="Neegalaĵo de triangulo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li> <li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eu.wikipedia.org/wiki/Desberdintza_triangeluar" title="Desberdintza triangeluar – Basque" lang="eu" hreflang="eu" data-title="Desberdintza triangeluar" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li> <li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fa.wikipedia.org/wiki/%25D9%2586%25D8%25A7%25D8%25A8%25D8%25B1%25D8%25A7%25D8%25A8%25D8%25B1%25DB%258C_%25D9%2585%25D8%25AB%25D9%2584%25D8%25AB%25DB%258C" title="نابرابری مثلثی – Persian" lang="fa" hreflang="fa" data-title="نابرابری مثلثی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li> <li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wikipedia.org/wiki/In%25C3%25A9galit%25C3%25A9_triangulaire" title="Inégalité triangulaire – French" lang="fr" hreflang="fr" data-title="Inégalité triangulaire" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li> <li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://gl.wikipedia.org/wiki/Desigualdade_triangular" title="Desigualdade triangular – Galician" lang="gl" hreflang="gl" data-title="Desigualdade triangular" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li> <li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ko.wikipedia.org/wiki/%25EC%2582%25BC%25EA%25B0%2581_%25EB%25B6%2580%25EB%2593%25B1%25EC%258B%259D" title="삼각 부등식 – Korean" lang="ko" hreflang="ko" data-title="삼각 부등식" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li> <li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hy.wikipedia.org/wiki/%25D4%25B5%25D5%25BC%25D5%25A1%25D5%25B6%25D5%25AF%25D5%25B5%25D5%25A1%25D5%25B6_%25D5%25A1%25D5%25B6%25D5%25B0%25D5%25A1%25D5%25BE%25D5%25A1%25D5%25BD%25D5%25A1%25D6%2580%25D5%25B8%25D6%2582%25D5%25A9%25D5%25B5%25D5%25B8%25D6%2582%25D5%25B6" title="Եռանկյան անհավասարություն – Armenian" lang="hy" hreflang="hy" data-title="Եռանկյան անհավասարություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li> <li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hi.wikipedia.org/wiki/%25E0%25A4%25A4%25E0%25A5%258D%25E0%25A4%25B0%25E0%25A4%25BF%25E0%25A4%25AD%25E0%25A5%2581%25E0%25A4%259C_%25E0%25A4%2585%25E0%25A4%25B8%25E0%25A4%25AE%25E0%25A4%25BF%25E0%25A4%2595%25E0%25A4%25BE" title="त्रिभुज असमिका – Hindi" lang="hi" hreflang="hi" data-title="त्रिभुज असमिका" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li> <li class="interlanguage-link interwiki-id mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://id.wikipedia.org/wiki/Pertidaksamaan_segitiga" title="Pertidaksamaan segitiga – Indonesian" lang="id" hreflang="id" data-title="Pertidaksamaan segitiga" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li> <li class="interlanguage-link interwiki-is mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://is.wikipedia.org/wiki/%25C3%259Er%25C3%25ADhyrnings%25C3%25B3jafna" title="Þríhyrningsójafna – Icelandic" lang="is" hreflang="is" data-title="Þríhyrningsójafna" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li> <li class="interlanguage-link interwiki-it mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://it.wikipedia.org/wiki/Disuguaglianza_triangolare" title="Disuguaglianza triangolare – Italian" lang="it" hreflang="it" data-title="Disuguaglianza triangolare" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li> <li class="interlanguage-link interwiki-he mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://he.wikipedia.org/wiki/%25D7%2590%25D7%2599-%25D7%25A9%25D7%2595%25D7%2595%25D7%2599%25D7%2595%25D7%259F_%25D7%2594%25D7%259E%25D7%25A9%25D7%2595%25D7%259C%25D7%25A9" title="אי-שוויון המשולש – Hebrew" lang="he" hreflang="he" data-title="אי-שוויון המשולש" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li> <li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lt.wikipedia.org/wiki/Trikampio_nelygyb%25C4%2597" title="Trikampio nelygybė – Lithuanian" lang="lt" hreflang="lt" data-title="Trikampio nelygybė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li> <li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lij.wikipedia.org/wiki/Disegualiansa_triangol%25C3%25A2" title="Disegualiansa triangolâ – Ligurian" lang="lij" hreflang="lij" data-title="Disegualiansa triangolâ" data-language-autonym="Ligure" data-language-local-name="Ligurian" class="interlanguage-link-target">Ligure</a></li> <li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hu.wikipedia.org/wiki/H%25C3%25A1romsz%25C3%25B6g-egyenl%25C5%2591tlens%25C3%25A9g" title="Háromszög-egyenlőtlenség – Hungarian" lang="hu" hreflang="hu" data-title="Háromszög-egyenlőtlenség" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li> <li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Driehoeksongelijkheid" title="Driehoeksongelijkheid – Dutch" lang="nl" hreflang="nl" data-title="Driehoeksongelijkheid" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li> <li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ja.wikipedia.org/wiki/%25E4%25B8%2589%25E8%25A7%2592%25E4%25B8%258D%25E7%25AD%2589%25E5%25BC%258F" title="三角不等式 – Japanese" lang="ja" hreflang="ja" data-title="三角不等式" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li> <li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nn.wikipedia.org/wiki/Trekantulikskapen" title="Trekantulikskapen – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Trekantulikskapen" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li> <li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pl.wikipedia.org/wiki/Nier%25C3%25B3wno%25C5%259B%25C4%2587_tr%25C3%25B3jk%25C4%2585ta" title="Nierówność trójkąta – Polish" lang="pl" hreflang="pl" data-title="Nierówność trójkąta" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li> <li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pt.wikipedia.org/wiki/Desigualdade_triangular" title="Desigualdade triangular – Portuguese" lang="pt" hreflang="pt" data-title="Desigualdade triangular" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li> <li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ro.wikipedia.org/wiki/Inegalitatea_triunghiului" title="Inegalitatea triunghiului – Romanian" lang="ro" hreflang="ro" data-title="Inegalitatea triunghiului" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li> <li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ru.wikipedia.org/wiki/%25D0%259D%25D0%25B5%25D1%2580%25D0%25B0%25D0%25B2%25D0%25B5%25D0%25BD%25D1%2581%25D1%2582%25D0%25B2%25D0%25BE_%25D1%2582%25D1%2580%25D0%25B5%25D1%2583%25D0%25B3%25D0%25BE%25D0%25BB%25D1%258C%25D0%25BD%25D0%25B8%25D0%25BA%25D0%25B0" title="Неравенство треугольника – Russian" lang="ru" hreflang="ru" data-title="Неравенство треугольника" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li> <li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ckb.wikipedia.org/wiki/%25D9%2584%25D8%25A7%25D8%25B3%25DB%2595%25D9%2586%25DA%25AF%25DB%2595%25DB%258C_%25D8%25B3%25DB%258E%25DA%25AF%25DB%2586%25D8%25B4%25DB%2595%25DB%258C%25DB%258C" title="لاسەنگەی سێگۆشەیی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="لاسەنگەی سێگۆشەیی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li> <li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sr.wikipedia.org/wiki/%25D0%259D%25D0%25B5%25D1%2598%25D0%25B5%25D0%25B4%25D0%25BD%25D0%25B0%25D0%25BA%25D0%25BE%25D1%2581%25D1%2582_%25D1%2582%25D1%2580%25D0%25BE%25D1%2583%25D0%25B3%25D0%25BB%25D0%25B0" title="Неједнакост троугла – Serbian" lang="sr" hreflang="sr" data-title="Неједнакост троугла" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li> <li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fi.wikipedia.org/wiki/Kolmioep%25C3%25A4yht%25C3%25A4l%25C3%25B6" title="Kolmioepäyhtälö – Finnish" lang="fi" hreflang="fi" data-title="Kolmioepäyhtälö" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li> <li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Triangelolikheten" title="Triangelolikheten – Swedish" lang="sv" hreflang="sv" data-title="Triangelolikheten" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li> <li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ta.wikipedia.org/wiki/%25E0%25AE%2585%25E0%25AE%259F%25E0%25AE%25BF%25E0%25AE%25AA%25E0%25AF%258D%25E0%25AE%25AA%25E0%25AE%259F%25E0%25AF%2588_%25E0%25AE%25AE%25E0%25AF%2581%25E0%25AE%2595%25E0%25AF%258D%25E0%25AE%2595%25E0%25AF%258B%25E0%25AE%25A3%25E0%25AE%259A%25E0%25AF%258D_%25E0%25AE%259A%25E0%25AE%25AE%25E0%25AE%25A9%25E0%25AE%25BF%25E0%25AE%25A9%25E0%25AF%258D%25E0%25AE%25AE%25E0%25AF%2588" title="அடிப்படை முக்கோணச் சமனின்மை – Tamil" lang="ta" hreflang="ta" data-title="அடிப்படை முக்கோணச் சமனின்மை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li> <li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tr.wikipedia.org/wiki/%25C3%259C%25C3%25A7gen_e%25C5%259Fitsizli%25C4%259Fi" title="Üçgen eşitsizliği – Turkish" lang="tr" hreflang="tr" data-title="Üçgen eşitsizliği" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li> <li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uk.wikipedia.org/wiki/%25D0%259D%25D0%25B5%25D1%2580%25D1%2596%25D0%25B2%25D0%25BD%25D1%2596%25D1%2581%25D1%2582%25D1%258C_%25D1%2582%25D1%2580%25D0%25B8%25D0%25BA%25D1%2583%25D1%2582%25D0%25BD%25D0%25B8%25D0%25BA%25D0%25B0" title="Нерівність трикутника – Ukrainian" lang="uk" hreflang="uk" data-title="Нерівність трикутника" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li> <li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://vi.wikipedia.org/wiki/B%25E1%25BA%25A5t_%25C4%2591%25E1%25BA%25B3ng_th%25E1%25BB%25A9c_tam_gi%25C3%25A1c" title="Bất đẳng thức tam giác – Vietnamese" lang="vi" hreflang="vi" data-title="Bất đẳng thức tam giác" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li> <li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh-yue.wikipedia.org/wiki/%25E4%25B8%2589%25E8%25A7%2592%25E4%25B8%258D%25E7%25AD%2589%25E5%25BC%258F" title="三角不等式 – Cantonese" lang="yue" hreflang="yue" data-title="三角不等式" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li> <li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh.wikipedia.org/wiki/%25E4%25B8%2589%25E8%25A7%2592%25E4%25B8%258D%25E7%25AD%2589%25E5%25BC%258F" title="三角不等式 – Chinese" lang="zh" hreflang="zh" data-title="三角不等式" data-language-autonym="中文" data-language-local-name="Chinese" 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Triangle inequality - Wikipedia