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Journal articles on the topic 'Triangle Inequality'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Hidayatin, Nur, and Frida Murtinasari. "Generalisasi Ketaksamaan Sinus pada Segitiga." Jurnal Axioma : Jurnal Matematika dan Pembelajaran 7, no. 1 (2022): 72–78. http://dx.doi.org/10.56013/axi.v7i1.1195.

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This study aims to find a generalization of the sine inequality of any triangles. This generalization is the general form of the sine inequality in a triangle when the angles given are not angles of the triangle, i.e. when the sum of the three angles is not equal to . The sine inequality that will be studied focuses on the inequalities of the sum and multiplication of sine in triangles. In the process, qualitative research methods are carried out in the form of literature review, namely studying the sum and the multiplication inequalities of sine in triangles which will then be developed and o

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2

Hoehn, Larry. "Geometrical Inequalities via Bisectors." Mathematics Teacher 82, no. 2 (1989): 96–99. http://dx.doi.org/10.5951/mt.82.2.0096.

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The proofs of several theorems in secondary school mathematics that involve geometrical inequalities are more complicated than they really need to be. This article presents an easier-to-understand alternative to the usual proofs of inequalities in triangles. The only inequality with which we need to assume familiarity is the triangle inequality (i.e., the sum of any two sides of a plane triangle is greater than the third side).

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3

Yang, Zheng, Lirong Jian, Chenshu Wu, and Yunhao Liu. "Beyond triangle inequality." ACM Transactions on Sensor Networks 9, no. 2 (2013): 1–20. http://dx.doi.org/10.1145/2422966.2422983.

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4

Shevchishin, Vsevolod, and Gleb Smirnov. "Symplectic triangle inequality." Proceedings of the American Mathematical Society 148, no. 4 (2020): 1389–97. http://dx.doi.org/10.1090/proc/14842.

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5

Greenhoe, D. "Properties of distance spaces with power triangle inequalities." Carpathian Mathematical Publications 8, no. 1 (2016): 51–82. http://dx.doi.org/10.15330/cmp.8.1.51-82.

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Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the "power triangle inequality". The pow

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6

Barnes, Benedict, E. D. J. O. Wusu-Ansah, S. K. Amponsah, and I. A. Adjei. "The Proofs of Triangle Inequality Using Binomial Inequalities." European Journal of Pure and Applied Mathematics 11, no. 1 (2018): 352–61. http://dx.doi.org/10.29020/nybg.ejpam.v11i1.3165.

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In this paper, we introduce the different ways of proving the triangle inequality ku − vk ≤ kuk + kvk, in the Hilbert space. Thus, we prove this triangle inequality through the binomial inequality and also, prove it through the Euclidean norm. The first generalized procedure for proving the triangle inequality is feasible for any even positive integer n. The second alternative proof of the triangle inequality establishes the Euclidean norm of any two vectors in the Hilbert space.

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7

Vu, Tuyen. "Spectral inequality for Dirac right triangles." Journal of Mathematical Physics 64, no. 4 (2023): 041502. http://dx.doi.org/10.1063/5.0147732.

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We consider a Dirac operator on right triangles, subject to infinite-mass boundary conditions. We conjecture that the lowest positive eigenvalue is minimized by the isosceles right triangle under the area or perimeter constraints. We prove this conjecture under extra geometric hypotheses relying on a recent approach of Briet and Krejčiřík [J. Math. Phys. 63, 013502 (2022)].

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8

Bencze, Mihaly, and GCHQ Problems Group. "A Triangle Inequality: 10644." American Mathematical Monthly 106, no. 5 (1999): 476. http://dx.doi.org/10.2307/2589167.

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9

Bailey, Herbert R., and Robert Bannister. "A Stronger Triangle Inequality." College Mathematics Journal 28, no. 3 (1997): 182. http://dx.doi.org/10.2307/2687521.

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10

Shahbari, Juhaina A., and Moshe Stupel. "106.13 A triangle inequality." Mathematical Gazette 106, no. 565 (2022): 138. http://dx.doi.org/10.1017/mag.2022.28.

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11

Bailey, Herbert R., and Robert Bannister. "A Stronger Triangle Inequality." College Mathematics Journal 28, no. 3 (1997): 182–86. http://dx.doi.org/10.1080/07468342.1997.11973859.

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12

Retkes, Zoltan. "Generalising a triangle inequality." Mathematical Gazette 102, no. 555 (2018): 422–27. http://dx.doi.org/10.1017/mag.2018.108.

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The main goal of this paper is to give a deeper understanding of the geometrical inequality proposed by Martin Lukarevski in [1]. In order to formulate our results we shall introduce and use the following notation throughout this paper. Let A1A2A3 be a triangle a1, a2, a3, the lengths of the sides opposite to A1, A2, A3 respectively, P an arbitrary inner point of it xi, the distance of P from the side of length ai. Let r, R be the inradius and circumradius of the triangle hi, the altitude belonging to side ai, Δ the area and finally let α be a real parameter. We adopt also the use of Σui to re

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13

Nánásiová, O. l̆ga, and L̆ubica Valášková. "Marginality and Triangle Inequality." International Journal of Theoretical Physics 49, no. 12 (2010): 3199–208. http://dx.doi.org/10.1007/s10773-010-0414-2.

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14

Fujii, Masatoshi, Mikio Kato, Kichi-Suke Saito, and Takayuki Tamura. "Sharp mean triangle inequality." Mathematical Inequalities & Applications, no. 4 (2010): 743–52. http://dx.doi.org/10.7153/mia-13-53.

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15

Nasyiitoh, Hikma Khilda. "KETAKSAMAAN JUMLAHAN SINUS PANGKAT 2^n YANG BERLAKU PADA SEGITIGA LANCIP." JURNAL SILOGISME : Kajian Ilmu Matematika dan Pembelajarannya 3, no. 1 (2018): 21. http://dx.doi.org/10.24269/js.v3i1.955.

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Let \alpha, \beta, \gamma are angles in acute triangle ABC and a,b,c are the length of the triangle. By using the sine of angles as the relationship between the length of triangle and the radius of the circle circumscribed about a plane triangle, will be proven the sum inequality of quadratic sine in acute triangle. Then, by using the quadratic sum inequality of the sides of triangle will be extended for the case of the sum inequality of sine of order 2^n in acute triangle.

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16

Zhukovskaya, Zukhra, Sergey Zhukovskiy, and Richik Sengupta. "On exact triangle inequalities in (q1, q2)-quasimetric spaces." Tambov University Reports. Series: Natural and Technical Sciences, no. 125 (2019): 33–38. http://dx.doi.org/10.20310/1810-0198-2019-24-125-33-38.

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For arbitrary (q1, q2) -quasimetric space, it is proved that there exists a function f, such that f -triangle inequality is more exact than any (q1, q2) -triangle inequality. It is shown that this function f is the least one in the set of all concave continuous functions g for which g -triangle inequality hold.

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17

Recio, Tomás, Carlos Ueno, and María Pilar Vélez. "A Computational Approach to the Perimeter-Area Inequality in a Triangle." Axioms 14, no. 1 (2025): 40. https://doi.org/10.3390/axioms14010040.

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This paper explores the application of automated reasoning tools, specifically those implemented in GeoGebra Discovery, to the perimeter-area inequality in triangles. Focusing on the computational complex and real algebraic geometry methods behind these tools, this study analyzes a geometric construction involving a triangle with arbitrary side lengths and area, investigating the automated derivation of the relationship between the area and perimeter of a triangle, and showing that only equilateral triangles satisfy the exact perimeter-area equality. The main contribution of this work is to de

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18

Ajne, Björn. "Additivity of Chain-Ladder Projections." ASTIN Bulletin 24, no. 2 (1994): 311–18. http://dx.doi.org/10.2143/ast.24.2.2005072.

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AbstractIn this paper some results are given on the addivity of chain-ladder projections. Given two claims development triangles, when do their chain-ladder projections add up to the projections of the combined triangle, that is the triangle being the element-wise sum of the two given triangles?Necessary and sufficient conditions for equality are given. These are of a fairly simply form and are directly connected to the ordinary chain-ladder calculations. In addition, sufficient conditions of the same form are given for inequality between the combined projection vector and the sum of the two o

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19

Kesavan, S. "From the triangle inequality to the isoperimetric inequality." Resonance 19, no. 2 (2014): 135–48. http://dx.doi.org/10.1007/s12045-014-0017-y.

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20

R., Elakkiya, and Panneer Selvam A. "A STUDY ON GEOMETRIC INEQUALITIES." International Journal of Current Research and Modern Education, Special Issue (August 15, 2017): 88–89. https://doi.org/10.5281/zenodo.843539.

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21

Satnoianu, Razvan, and Walther Janous. "A Two-Triangle Inequality: 11022." American Mathematical Monthly 112, no. 3 (2005): 280. http://dx.doi.org/10.2307/30037460.

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22

Saito, Kichi-Suke, Runling An, Hiroyasu Mizuguchi, and Ken-Ichi Mitani. "Another Aspect of Triangle Inequality." ISRN Mathematical Analysis 2011 (April 14, 2011): 1–5. http://dx.doi.org/10.5402/2011/514184.

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We introduce the notion of ψ-norm by considering the fact that an absolute normalized norm on C2 corresponds to a continuous convex function ψ on the unit interval [0,1] with some conditions. This is a generalization of the notion of q-norm introduced by Belbachir et al. (2006). Then we show that a ψ-norm is a norm in the usual sense.

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23

Dees, Marco. "Maudlin on the Triangle Inequality." Thought: A Journal of Philosophy 4, no. 2 (2015): 124–30. http://dx.doi.org/10.1002/tht3.165.

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24

Harrison, Elizabeth P. "Sharing Teaching Ideas: Using the Law of Cosines to Teach the Ambiguous Case of the Law of Sines." Mathematics Teacher 95, no. 2 (2002): 114–16. http://dx.doi.org/10.5951/mt.95.2.0114.

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The laws of sines and cosines easily lend themselves to links with other areas of algebra and geometry. The most-used link is probably that of congruent triangles, but additional links exist with imaginary numbers, the quadratic formula, parabolas, zeros of functions, and the triangle inequality.

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25

徐, 小伟. "A New Inequality Derivates from a Classical Triangle Inequality." Pure Mathematics 04, no. 01 (2014): 21–26. http://dx.doi.org/10.12677/pm.2014.41004.

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26

Wu, Yuwei, and Gregory S. Chirikjian. "Trace inequalities and kinematic metrics." Robotica 42, no. 12 (2024): 4103–19. https://doi.org/10.1017/s0263574724000778.

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AbstractKinematics remains one of the cornerstones of robotics, and over the decade, Robotica has been one of the venues in which groundbreaking work in kinematics has always been welcome. A number of works in the kinematics community have addressed metrics for rigid-body motions in multiple different venues. An essential feature of any distance metric is the triangle inequality. Here, relationships between the triangle inequality for kinematic metrics and so-called trace inequalities are established. In particular, we show that the Golden-Thompson inequality (a particular trace inequality fro

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27

Aglić Aljinović, Andrea, Ilko Brnetić, and Ana Žgaljić Keko. "Triangle inequality for quantum integral operator." Acta mathematica Spalatensia 2 (December 1, 2022): 97–110. http://dx.doi.org/10.32817/ams.2.7.

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We show that general integral triangle inequality does not hold for shifted q-integrals. Furthermore, we obtain a triangle inequality for shifted qintegrals. We also give an estimate for q-integrable product and use it to refine some recently obtained Ostrowski inequalities for quantum calculus.

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28

Chen, Chao-Ping, and Feng Qi. "Note on Alzer's inequality." Tamkang Journal of Mathematics 37, no. 1 (2006): 11–14. http://dx.doi.org/10.5556/j.tkjm.37.2006.175.

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29

Tran, Quang Hung. "Some generalisations of Weitzenböck’s inequality." Mathematical Gazette 107, no. 570 (2023): 405–11. http://dx.doi.org/10.1017/mag.2023.92.

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Throughout this Article, we use the following notations for the triangle ABCa, b and c are the lengths of the sides BC, CA and AB, respectively,Δ denotes the area of triangle ABC,ha, hb and hc are the lengths of the altitudes through the vertices A, B and C, respectively,ma, mb and mc are the lengths of the medians through the vertices A, B and C, respectively.

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30

Melville, John, Jason Heller, and Stony Brook. "A Classic Pointed Triangle Inequality: 10965." American Mathematical Monthly 111, no. 8 (2004): 726. http://dx.doi.org/10.2307/4145057.

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31

Harte, Robin. "The triangle inequality in C*algebras." Filomat 20, no. 2 (2006): 51–53. http://dx.doi.org/10.2298/fil0602053h.

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32

Jäkel, Frank, Bernhard Schölkopf, and Felix A. Wichmann. "Similarity, kernels, and the triangle inequality." Journal of Mathematical Psychology 52, no. 5 (2008): 297–303. http://dx.doi.org/10.1016/j.jmp.2008.03.001.

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33

Barbu, Cătălin, and Laurian-Ioan Pişcoran. "Andrica-Iwata's inequality in hyperbolic triangle." Mathematical Inequalities & Applications, no. 3 (2012): 631–37. http://dx.doi.org/10.7153/mia-15-55.

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34

Rudnicki, Łukasz, Zbigniew Puchała, Paweł Horodecki, and Karol Życzkowski. "Constructive entanglement test from triangle inequality." Journal of Physics A: Mathematical and Theoretical 47, no. 42 (2014): 424035. http://dx.doi.org/10.1088/1751-8113/47/42/424035.

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35

Petrov, Evgeniy, and Ruslan Salimov. "A note on generalized four-point inequality." Ukrainian Mathematical Bulletin 20, no. 1 (2023): 107–23. http://dx.doi.org/10.37069/1810-3200-2023-20-1-6.

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In 2017, M. Bessenyei and Z. Páles [1] introduced a definition of a triangle function that generates a concept of a generalized triangle inequality in semimetric spaces. Inspired by this concept, we discuss already known inequalities in metric spaces that relate six distances determined by four points and introduce a definition of the generalized four-point inequality in semimetric spaces. Conditions under which quasisymmetric mappings and quasi-Möbius mappings between semimetric spaces preserve such an inequality are obtained.

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36

Volenec, Vladimir, Zdenka Kolar-Begović, and Ružica Kolar-Šuper. "Heptagonal triangle as the extreme triangle of Dixmier-Kahane-Nicolas inequality." Mathematical Inequalities & Applications, no. 4 (2009): 773–79. http://dx.doi.org/10.7153/mia-12-60.

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37

Verulava, Tengiz. "Economic growth, inequality and poverty: policy issues and challenges." Economic Profile 20, no. 1(29) (2025): 32–41. https://doi.org/10.52244/ep.2025.29.03.

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Economic growth, inequality and poverty are closely linked. Although economic growth can contribute to the increase in general well-being, increasing inequality often reduces the effectiveness of economic progress and increases the level of poverty. The triangle model (poverty-growth-inequality triangle) developed by Bourguignon (2004) remains one of the most important theories in the analysis of developing economies. Its main advantage is the understanding of the interaction of growth and inequality as a determining factor in the dynamics of poverty. Economic growth is not enough to eliminate

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38

KRIVELEVICH, MICHAEL. "Triangle Factors in Random Graphs." Combinatorics, Probability and Computing 6, no. 3 (1997): 337–47. http://dx.doi.org/10.1017/s0963548397003106.

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For a graph G=(V, E) on n vertices, where 3 divides n, a triangle factor is a subgraph of G, consisting of n/3 vertex disjoint triangles (complete graphs on three vertices). We discuss the problem of determining the minimal probability p=p(n), for which a random graph G∈[Gscr ](n, p) contains almost surely a triangle factor. This problem (in a more general setting) has been studied by Alon and Yuster and by Ruciński, their approach implies p=O((log n/n)1/2). Our main result is that p=O(n)−3/5) already suffices. The proof is based on a multiple use of the Janson inequality. Our approach can be

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39

Moszyńska, Maria, and Wolf-Dieter Richter. "Reverse triangle inequality. Antinorms and semi-antinorms." Studia Scientiarum Mathematicarum Hungarica 49, no. 1 (2012): 120–38. http://dx.doi.org/10.1556/sscmath.49.2012.1.1192.

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The paper concerns a biunique correspondence between some positively homogeneous functions on ℝn and some star-shaped sets with nonempty interior, symmetric with respect to the origin (Theorems 3.5 and 4.3).

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40

Liu, Jian. "A Pedal Triangle Inequality with the Exponents." International Journal of Open Problems in Computer Science and Mathematics 5, no. 4 (2012): 16–24. http://dx.doi.org/10.12816/0006135.

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41

Martín, José V., and Ángel Plaza. "A Triangle Inequality and its Elementary Proof." Math Horizons 15, no. 4 (2008): 30. http://dx.doi.org/10.1080/10724117.2008.11974774.

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42

Nelsen, Roger B. "Euler's Triangle Inequality via Proofs Without Words." Mathematics Magazine 81, no. 1 (2008): 58–61. http://dx.doi.org/10.1080/0025570x.2008.11953529.

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43

Sawhney, Mehtaab. "An Unusual Proof of the Triangle Inequality." College Mathematics Journal 49, no. 3 (2018): 218. http://dx.doi.org/10.1080/07468342.2017.1397993.

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44

Gourvès, Laurent, Jérôme Monnot, Fanny Pascual, and Daniel Vanderpooten. "Bi-objective matchings with the triangle inequality." Theoretical Computer Science 670 (March 2017): 1–10. http://dx.doi.org/10.1016/j.tcs.2017.01.012.

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45

Yearsley, James M., Albert Barque-Duran, Elisa Scerrati, James A. Hampton, and Emmanuel M. Pothos. "The triangle inequality constraint in similarity judgments." Progress in Biophysics and Molecular Biology 130 (November 2017): 26–32. http://dx.doi.org/10.1016/j.pbiomolbio.2017.03.005.

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46

Gu, Xinhua, and Pui Sun Tam. "The saving–growth–inequality triangle in China." Economic Modelling 33 (July 2013): 850–57. http://dx.doi.org/10.1016/j.econmod.2013.06.001.

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47

Kumari, Panda, Badriah Alamri, Nawab Hussain, and Sumit Chandok. "Unification of the Fixed Point in Integral Type Metric Spaces." Symmetry 10, no. 12 (2018): 732. http://dx.doi.org/10.3390/sym10120732.

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In metric fixed point theory, the conditions like “symmetry” and “triangle inequality” play a predominant role. In this paper, we introduce a new kind of metric space by using symmetry, triangle inequality, and other conditions like self-distances are zero. In this paper, we introduce the weaker forms of integral type metric spaces, thereby we establish the existence of unique fixed point theorems. As usual, illustrations and counter examples are provided wherever necessary.

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48

Ayoob, Irshad, Ng Zhen Chuan, and Nabil Mlaiki. "Double-Composed Metric Spaces." Mathematics 11, no. 8 (2023): 1866. http://dx.doi.org/10.3390/math11081866.

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The double-controlled metric-type space (X,D) is a metric space in which the triangle inequality has the form D(η,μ)≤ζ1(η,θ)D(η,θ)+ζ2(θ,μ)D(θ,μ) for all η,θ,μ∈X. The maps ζ1,ζ2:X×X→[1,∞) are called control functions. In this paper, we introduce a novel generalization of a metric space called a double-composed metric space, where the triangle inequality has the form D(η,μ)≤αD(η,θ)+βD(θ,μ) for all η,θ,μ∈X. In our new space, the control functions α,β:[0,∞)→[0,∞) are composed of the metric D in the triangle inequality, where the control functions ζ1,ζ2:X×X→[1,∞) in a double-controlled metric-type

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49

WU, YU-DONG, ZHI-HUA ZHANG, and ZHI-GANG WANG. "On Edwards–Child’s inequality." Creative Mathematics and Informatics 20, no. 1 (2011): 96–105. http://dx.doi.org/10.37193/cmi.2011.01.11.

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In this paper, by making use of one of Chen’s theorems and the method of mathematical analysis with the computer software Maple (Version 9.0), we refine Edwards–Child’s inequality and solve the conjecture ... involving the semi-perimeter p, the circumradius R and the inradius r of the triangle, which was posed by Liu.

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50

Maligranda, Lech. "Some remarks on the triangle inequality for norms." Banach Journal of Mathematical Analysis 2, no. 2 (2008): 31–41. http://dx.doi.org/10.15352/bjma/1240336290.

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