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{{Short description|Triangle comparison theorem in Riemannian geometry}} ...'Toponogov's theorem''' (named after [[Victor Andreevich Toponogov]]) is a triangle comparison theorem. ...

...bn=978-0-387-94826-3 }}</ref> and is so named for its resemblance to the [[triangle inequality]]. It is an important lemma in the proof of the [[Plünnecke-Ruzs ...ath>A-B</math> denotes <math>\{a-b:a\in A,b\in B\}</math>. Then, the Ruzsa triangle inequality states the following. ...

...triangle's angles must sum to 180° in [[Euclidean geometry]], no Euclidean triangle can have more than one obtuse angle. ...different types of '''oblique triangles'''—triangles that are not [[right triangle]]s because they do not have any [[right angle]]s (90°). ...

{{short description|Triangle with circular arc edges}} In [[geometry]], a '''circular triangle''' is a [[triangle]] with circular [[Arc (geometry)|arc]] [[edge (geometry)|edges]]. ...

...erwise estimating the modulus of the [[exponential sum]] by means of the [[triangle inequality]] as <math>\scriptstyle\le\, N</math> provides a better bound. [[Category:Inequalities]] ...

...proportion 13:17:7, its three [[Median (geometry)|medians]] (brown), and a triangle [[Similarity (geometry)|similar]] to the original one whose sides are trans ...may be [[translation (geometry)|translated]] to form the sides of a second triangle that is [[Similarity (geometry)|similar]] to the first one. ...

...ngruence. A triangle is said to be '''5-Con capable''' if there is another triangle which is almost congruent to it. ...parts (sides, angles) of one triangle congruent to five parts of the other triangle, but <u>not</u> be congruent triangles.</blockquote> ...

...be used to solve [[integer programming]] problems for the given system of inequalities, a problem that can otherwise be more difficult. ...olytope]] of any [[partially ordered set]], a polytope defined by pairwise inequalities between coordinates corresponding to comparable elements in the set.{{r|s}} ...

[[File:NeumannTriangle.svg|thumb|Figure 4: Neumann's triangle relating the surface energies and contact angles of three fluid phases coex ...og to a triangle known as Neumann’s triangle, shown in Figure 4. Neumann’s triangle is consistent with the geometrical restriction that <math>\alpha+\beta+\the ...

===Ruzsa triangle inequality=== {{main|Ruzsa triangle inequality}} <!-- The referenced article on the Ruzsa triangle inequality is currently a draft and has been submitted concurrently with th ...

...urnal|last=Hatami, H., Norine, S.|date=2011|title=Undecidability of linear inequalities in graph homomorphism densities.|url=https://www.ams.org/journals/jams/2011 For example, the triangle density in a graphon is given by ...

...to a branch of [[extremal graph theory]] concerning [[Homomorphism density|inequalities in homomorphism densities]]. Roughly speaking, <math>F</math> is a common [ ...e closely related to other graph notions dealing with homomorphism density inequalities. For example, common graphs are a more general case of [[Sidorenko's conjec ...

Below are three examples of incidence matrices: the [[triangle graph]] (a cycle of length 3), a square graph (a cycle of length 4), and th ...point in this polytope is a [[fractional matching]]. For example, in the [[triangle graph]] there are 3 edges, and the corresponding linear program has the fol ...

...piral of Theodorus.svg|thumb|right|400px|The spiral of Theodorus up to the triangle with a hypotenuse of <math>\sqrt{17}</math>]] ...r '''Pythagoras's snail'''){{r|KAHN2}} is a [[spiral]] composed of [[right triangle]]s, placed edge-to-edge. It was named after [[Theodorus of Cyrene]]. ...

...and ''{{mvar|c}}'' are the lengths of the sides of a [[triangle]] then the triangle inequality states that ...th zero [[area]]. In [[Euclidean geometry]] and some other geometries, the triangle inequality is a theorem about vectors and vector lengths ([[norm (mathemati ...

...h>-plane with <math>0\le x\le y \le 1</math>. This is an [[isosceles right triangle]] with vertices at (0,0), (0,1), and (1,1). *Inequalities <math>0\le f(x)</math> for each minimal element <math>x</math> of the parti ...

Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertic ...ious types of segment), as well as [[list of triangle inequalities|various inequalities]]. ...

==Inequalities== *The [[triangle inequality]]: <math>\|\mathbf{A + B}\| \le \| \mathbf{A}\| + \|\mathbf{B}\| ...

* the first proof of the existence of the [[orthocentre]] of a triangle, ...theorem in geometry|distance between the incentre and circumcentre]] of a triangle, ...

In [[mathematics]], '''systolic inequalities for curves on surfaces''' were first studied by [[Charles Loewner]] in 1949 ...al congruence subgroups of the [[(2,3,7) triangle group|(2,3,7) hyperbolic triangle group]] satisfy the bound ...