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Journal articles on the topic 'Triangle'

Author: Grafiati

Published: 4 June 2021

Last updated: 29 July 2024

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1

Mui, Grace, and Jennifer Mailley. "A tale of two triangles: comparing the Fraud Triangle with criminology’s Crime Triangle." Accounting Research Journal 28, no. 1 (July 6, 2015): 45–58. http://dx.doi.org/10.1108/arj-10-2014-0092.

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Purpose – This paper aims to propose the application of the Crime Triangle of Routine Activity Theory to fraud events as a complement to the universally accepted Fraud Triangle. Design/methodology/approach – The application of the Crime Triangle is illustrated using scenarios of asset misappropriations by type of perpetrator: external perpetrator, employee, management and the board and its governing bodies. Findings – The Crime Triangle complements the Fraud Triangle’s perpetrator-centric focus by examining the environment where fraud occurs and the relevant parties that play their role in preventing fraud or not playing their role, and thus, allowing the occurrence of fraud. Applying both triangles to a fraud event provides a comprehensive view of the fraud event. Research limitations/implications – The scenarios are limited to asset misappropriations with one perpetrator. Future research can apply both triangles to different types of fraud and cases where perpetrators collude to commit fraud. Practical implications – This paper maps the Crime Triangle to the Fraud Triangle to provide forensic accounting practitioners and researchers with a comprehensive perspective of a fraud event. This comprehensive perspective of fraud is the starting point to designing fraud risk management strategies that address both the perpetrator and the environment where the fraud event occurs. Originality/value – This paper is the first to propose the application of the established Crime Triangle environmental criminology theory as a complement to the Fraud Triangle to obtain a comprehensive perspective of a fraud event.

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2

Odehnal, Boris. "Two Convergent Triangle Tunnels." KoG, no. 22 (2018): 3–11. http://dx.doi.org/10.31896/k.22.1.

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A semi-orthogonal path is a polygon inscribed into a given polygon such that the $i$-th side of the path is orthogonal to the $i$-th side of the given polygon. Especially in the case of triangles, the closed semi-orthogonal paths are triangles which turn out to be similar to the given triangle. The iteration of the construction of semi-orthogonal paths in triangles yields infinite sequences of nested and similar triangles. We show that these two different sequences converge towards the bicentric pair of the triangle's Brocard points. Furthermore, the relation to discrete logarithmic spirals allows us to give a very simple, elementary, and new constructions of the sequences' limits, the Brocard points. We also add some remarks on semi-orthogonal paths in non-Euclidean geometries and in $n$-gons.

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3

Hikmatovich, Ibragimov Husniddin. "Connection Between A Right Triangle And An Equal Side Triangle." American Journal of Interdisciplinary Innovations and Research 02, no. 11 (November 30, 2020): 105–14. http://dx.doi.org/10.37547/tajiir/volume02issue11-20.

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There is some evidence that a right triangle and an equilateral triangle are related. Information about Pythagorean numbers is given. The geometric meaning of the relationship between right triangles and equilateral triangles is shown. The geometric meaning of the relationship between an equilateral triangle and an equilateral triangle is shown.

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4

Jurkin, Ema. "Poncelet Porisms and Loci of Centers in the Isotropic Plane." Mathematics 12, no. 4 (February 19, 2024): 618. http://dx.doi.org/10.3390/math12040618.

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Any triangle in an isotropic plane has a circumcircle u and incircle i. It turns out that there are infinitely many triangles with the same circumcircle u and incircle i. This one-parameter family of triangles is called a poristic system of triangles. We study the trace of the centroid, the Feuerbach point, the symmedian point, the Gergonne point, the Steiner point and the Brocard points for such a system of triangles. We also study the traces of some further points associated with the triangles of the poristic family, and we prove that the vertices of the contact triangle, tangential triangle and anticomplementary triangle move on circles while the initial triangle traverses the poristic family.

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Wysiadecki, Grzegorz, Maciej Radek, R. Shane Tubbs, Joe Iwanaga, Jerzy Walocha, Piotr Brzeziński, and Michał Polguj. "Microsurgical Anatomy of the Inferomedial Paraclival Triangle: Contents, Topographical Relationships and Anatomical Variations." Brain Sciences 11, no. 5 (May 4, 2021): 596. http://dx.doi.org/10.3390/brainsci11050596.

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The inferomedial triangle is one of the two surgical triangles in the paraclival subregion of the skull base. It is delineated by the posterior clinoid process, the dural entrance of the trochlear nerve and the dural entrance of the abducens nerve. The aim of the present article is to describe the anatomical variations within the inferomedial triangle. Measurements of the triangle’s borders and area were supplemented by detailed observations of the topographical anatomy and various arrangements of its contents. Nine adult cadaveric heads (18 sides) and 28 sagittal head sections were studied. The mean area of the inferomedial triangle was estimated to be 60.7 mm2. The mean lengths of its medial, lateral and superior borders were 16.1 mm, 11.9 mm and 10.4 mm, respectively. The dorsal meningeal artery was identified within the inferomedial triangle in 37 out of 46 sides (80.4%). A well-developed petrosphenoidal ligament of Grüber was identified within the triangle on 36 sides (78.3%). Although some structures were variable, the constant contents of the inferomedial triangle were the posterior petroclinoid dural fold, the upper end of the petroclival suture, the gulfar segment of the abducens nerve and the posterior genu of the intracavernous internal carotid artery.

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6

Čerin, Zvonko. "On Napoleon triangles and propeller theorems." Mathematical Gazette 87, no. 508 (March 2003): 42–50. http://dx.doi.org/10.1017/s0025557200172092.

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In this paper we shall consider two situations in triangle geometry when equilateral triangles appear and then show that they are closely related.In the first (known as the Napoleon theorem) equilateral triangles BCAT, CABT, and ABCT, are built on the sides of an arbitrary triangle ABC and their centroids are (almost always) vertices of an equilateral triangle ANBNCN (known as a Napoleon triangle of ABC; see Figure 1).

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Kodrnja, Iva, and Helena Koncul. "The Loci of Vertices of Nedian Triangles." KoG, no. 21 (2017): 19–25. http://dx.doi.org/10.31896/k.21.5.

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In this article we observe nedians and nedian triangles of ratio $\eta$ for a given triangle. The locus of vertices of the nedian triangles for $\eta\in\mathbb{R}$ is found and its correlation with isotomic conjugates of the given triangle is shown. Furthermore, the curve on which lie vertices of a nedian triangle for fixed $\eta$, when we iterate nedian triangles, is found.

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8

ISMAIL, SHAHRINA. "PERFECT TRIANGLES ON THE CURVE." Journal of the Australian Mathematical Society 109, no. 1 (October 9, 2019): 68–80. http://dx.doi.org/10.1017/s144678871900003x.

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A Heron triangle is a triangle that has three rational sides $(a,b,c)$ and a rational area, whereas a perfect triangle is a Heron triangle that has three rational medians $(k,l,m)$. Finding a perfect triangle was stated as an open problem by Richard Guy [Unsolved Problems in Number Theory (Springer, New York, 1981)]. Heron triangles with two rational medians are parametrized by the eight curves $C_{1},\ldots ,C_{8}$ mentioned in Buchholz and Rathbun [‘An infinite set of heron triangles with two rational medians’, Amer. Math. Monthly 104(2) (1997), 106–115; ‘Heron triangles and elliptic curves’, Bull. Aust. Math.Soc. 58 (1998), 411–421] and Bácskái et al. [Symmetries of triangles with two rational medians, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.65.6533, 2003]. In this paper, we reveal results on the curve $C_{4}$ which has the property of satisfying conditions such that six of seven parameters given by three sides, two medians and area are rational. Our aim is to perform an extensive search to prove the nonexistence of a perfect triangle arising from this curve.

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9

Hidayatin, Nur, and Frida Murtinasari. "Generalisasi Ketaksamaan Sinus pada Segitiga." Jurnal Axioma : Jurnal Matematika dan Pembelajaran 7, no. 1 (June 6, 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 obtained new generalizations from the previous inequalities, namely generalizations of sine inequalities in triangles. These generalizations include generalizing the inequalities of the sum of sine in triangles and generalizing the inequalities of multiplication sine in triangles. To study this, it is necessary to first study the concepts of trigonometry, namely the definition of sine and cosine, the rules of sine and cosine; the relationship of the sine cosine to the sides of the triangle; the relationship of the radius of the circumcircle of the triangle to the sides and angles of the triangle; and arithmetic and geometric mean inequalities.The results of this study obtained the generalization of the sine inequality of any triangles. Keywords: inequality, sine, triangle

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10

Parry, C. F., and Clark Kimberling. "Triangle Centers and Central Triangles." Mathematical Gazette 85, no. 502 (March 2001): 172. http://dx.doi.org/10.2307/3620531.

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Leversha, Gerry, and Clark Kimberling. "Triangle Centers and Central Triangles." Mathematical Gazette 85, no. 502 (March 2001): 173. http://dx.doi.org/10.2307/3620532.

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12

Delahaye, Jean-Paul. "Découper un triangle en triangles." Pour la Science N° 500 - juin, no. 6 (January 6, 2019): 82–87. http://dx.doi.org/10.3917/pls.500.0082.

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13

Montoro, Carlos, and Regine Hampel. "CALL: A triangle of triangles." EuroCALL Review 20, no. 1 (March 22, 2012): 126. http://dx.doi.org/10.4995/eurocall.2012.16204.

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Institutional investment in technology and infrastructure for the provision of new online and self-access language learning opportunities is not always accompanied by the necessary changes in the practices of learners, practitioners and managers in higher education (Wertsch, 2002). As a result, feelings of frustration, helplessness and confusion in individuals can begin to emerge soon after the initial excitement and novelty wear off. A large investment project that has little or no overall impact may give rise to questions about the adequate use of resources and ultimately lead to abandoning, discontinuing or replacing the project. What lies at the root of failures to implement CALL institutionally? How can new practices emerge from existing ones? To explore these questions, in this paper we describe a project based on the assumption that the proposed CALL triangle consisting of the student, the teacher and the institution should be seen as a triangle of triangles, that is, the combination of three separate triangles, each one representing the specific activity system of the student, the teacher and the institution. The rationale is that these activities have different objects and motives as well as their own inner contradictions manifested in various ways. Building upon activity theory (Leontiev, 1978; Vygotsky 1987; Engeström, 1987) and expansive learning theory (Engeström, 1987) principles, the authors advocate the Change Laboratory methodology developed at CRADLE (University of Helsinki’s Center for Research on Activity, Development and Learning) to bridge gaps between the three activities of the student, the teacher and the institution by finding a shared object, building a common zone of proximal development and creating the necessary tools that could lead to the formation of a new collective activity.

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Garstang, R. H. "70.38 Triangles in a Triangle." Mathematical Gazette 70, no. 454 (December 1986): 288. http://dx.doi.org/10.2307/3616187.

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15

Dolan, Stan. "Triangles around a given triangle." Mathematical Gazette 99, no. 546 (November 2015): 432–43. http://dx.doi.org/10.1017/mag.2015.80.

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16

Lester, June A. "Triangles III: Complex triangle functions." Aequationes Mathematicae 53, no. 1-2 (February 1997): 4–35. http://dx.doi.org/10.1007/bf02215963.

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Lester, June A. "Triangles II: Complex triangle coordinates." Aequationes Mathematicae 52, no. 1 (February 1996): 215–45. http://dx.doi.org/10.1007/bf01818341.

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18

ZHANG, XIN-MIN, L. RICHARD HITT, BIN WANG, and JIU DING. "SIERPIŃSKI PEDAL TRIANGLES." Fractals 16, no. 02 (June 2008): 141–50. http://dx.doi.org/10.1142/s0218348x08003934.

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We generalize the construction of the ordinary Sierpiński triangle to obtain a two-parameter family of fractals we call Sierpiński pedal triangles. These fractals are obtained from a given triangle by recursively deleting the associated pedal triangles in a manner analogous to the construction of the ordinary Sierpiński triangle, but their fractal dimensions depend on the choice of the initial triangles. In this paper, we discuss the fractal dimensions of the Sierpiński pedal triangles and the related area ratio problem, and provide some computer-generated graphs of the fractals.

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19

Volenec, Vladimir, Zdenka Kolar-Begović, and Ružica Kolar-Šuper. "On Some Properties of the First Brocard Triangle in the Isotropic Plane." Mathematics 10, no. 9 (April 20, 2022): 1381. http://dx.doi.org/10.3390/math10091381.

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In this paper we introduce the first Brocard triangle of an allowable triangle in the isotropic plane and derive the coordinates of its vertices in the case of a standard triangle. We prove that the first Brocard triangle is homologous to the given triangle and that these two triangles are parallelogic. We consider the relationships between the first Brocard triangle and the Steiner axis, the Steiner point, and the Kiepert parabola of the triangle. We also investigate some other interesting properties of this triangle and consider relationships between the Euclidean and the isotropic case.

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20

SZILARD, ANDRAS, and SANDOR NAGYDOBAI KISS. "An area formula for the triangle of residual centroids and its generalizations." Creative Mathematics and Informatics 30, no. 1 (February 15, 2021): 11–18. http://dx.doi.org/10.37193/cmi.2021.01.02.

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In this paper we consider an inscribed triangle XY Z to a triangle ABC and we establish a relation between the area of these two triangles and the area of the triangle determined by the centroids of the residual triangles AZY, BXZ and CY X. Moreover we generalize this relation to the case of a general barycenter instead of centroid and also to quadrilaterals.

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Su, Xuzhong, and Xinjin Liu. "Theoretical research of spinning triangle division on spun yarn torque." International Journal of Clothing Science and Technology 31, no. 6 (November 4, 2019): 839–55. http://dx.doi.org/10.1108/ijcst-01-2019-0004.

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Purpose The purpose of this paper is to theoretically study the effects of ring spinning triangle division on spun yarn torques. Design/methodology/approach The case that the spinning triangle is divided into two parts, primary triangles and final triangle, is investigated. Theoretical model of yarn torque was given by linking the fiber tension in the spinning triangle to yarn torque under the assumption that the arrangement of fibers (substrands) in the substrands (yarn) is hexagonal close packing. Then, as an application of the proposed method, 14.6tex cotton yarns were taken as an example for the numerical simulations. Findings The fiber tensions in the divided spinning triangles and corresponding yarn torques were simulated numerically by using MATLAB software. The effects of division proportions and number of primary triangles on spun yarn torques are analyzed theoretically. Originality/value It is shown that suitable spinning triangle division is benefit for reducing yarn torque.

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22

Rodríguez-Hernández, Ana, and Michael T. Lawton. "Anatomical triangles defining surgical routes to posterior inferior cerebellar artery aneurysms." Journal of Neurosurgery 114, no. 4 (April 2011): 1088–94. http://dx.doi.org/10.3171/2010.8.jns10759.

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Object Surgical routes to posterior inferior cerebellar artery (PICA) aneurysms are opened between the vagus (cranial nerve [CN] X), accessory (CN XI), and hypoglossal (CN XII) nerves for safe clipping, but these routes have not been systematically defined. The authors describe 3 anatomical triangles and their relationships with PICA aneurysms, routes for surgical clipping, outcomes, and angiographically demonstrated anatomy. Methods The vagoaccesory triangle is defined by CN X superiorly, CN XI laterally, and the medulla medially. It is divided by CN XII into the suprahypoglossal triangle (above CN XII) and the infrahypoglossal triangle (below CN XII). From a consecutive surgical series of 71 PICA aneurysms in 70 patients, 51 aneurysms were analyzed using intraoperative photographs. Results Forty-three PICA aneurysms were located inside the vagoaccessory triangle and 8 were outside. Of the aneurysms inside the vagoaccessory triangle, 22 (51%) were exposed through the suprahypoglossal triangle and 19 (44%) through the infrahypoglossal triangle; 2 were between triangles. The lesions were evenly distributed between the anterior medullary (16 aneurysms), lateral medullary (19 aneurysms), and tonsillomedullary zones (16 aneurysms). Neurological and CN morbidity linked to aneurysms in the suprahypoglossal triangle was similar to that associated with aneurysms in the infrahypoglossal triangle, but no morbidity was associated with PICA aneurysms outside the vagoaccessory triangle. A distal PICA origin on angiography localized the aneurysm to the suprahypoglossal triangle in 71% of patients, and distal PICA aneurysms were localized to the infrahypoglossal triangle or outside the vagoaccessory triangle in 78% of patients. Conclusions The anatomical triangles and zones clarify the borders of operative corridors to PICA aneurysms and define the depth of dissection through the CNs. Deep dissection to aneurysms in the anterior medullary zone traverses CNs X, XI, and XII, whereas shallow dissection to aneurysms in the lateral medullary zone traverses CNs X and XI. Posterior inferior cerebellar artery aneurysms outside the vagoaccessory triangle are frequently distal and superficial to the lower CNs, and associated surgical morbidity is minimal. Angiography may preoperatively localize a PICA aneurysm's triangular anatomy based on the distal PICA origin or distal aneurysm location.

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Read, Emrys. "On the class of an integer triangle." Mathematical Gazette 106, no. 566 (June 22, 2022): 291–99. http://dx.doi.org/10.1017/mag.2022.69.

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Any mathematics student who has ever used the cosine rule to investigate simple properties of an integer triangle will immediately have realised that the cosine of each angle of the triangle must be a rational number. It is clear, however, that the same is not in general true for the sines. In [1], it is shown how to use a property of the sines of the angles of an integer triangle to categorise the triangle as being of a particular class. In this article, we develop some of the concepts and results of [1] to derive a method for generating integer triangles of a given class. Finally, we apply our results to find all primitive integer triangles in the particular case of Heronian triangles.

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Rešić, Sead, Alma Šehanović, and Amila Osmić. "ISOSCELES TRIANGLES ON THE SIDES OF A TRIANGLE." Journal Human Research in Rehabilitation 9, no. 1 (April 2019): 123–30. http://dx.doi.org/10.21554/hrr.041915.

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Famous construction of Fermat-Toricelly point of a triangle leads to the question is there a similar way to construct other isogonic centers of a triangle in a similar way. For a purpose we remember that Fermat-Torricelli point of a triangle ΔABC is obtained by constructing equilateral triangles outwardly on the sides AB,BC and CA. If we denote thirth vertices of those triangles by C1 ,A1 and B1 respectively, then the lines AA1 ,BB1 and CC1 concurr at the Fermat-Torricelli point of a triangle ΔABC (Van Lamoen, 2003). In this work we present the condition for the concurrence, of the lines AA1 ,BB1 and C1 , where C1 ,A1 and B1 are the vertices of an isosceles triangles constructed on the sides AB,BC and CA (not necessarily outwordly) of a triangle ΔABC. The angles at this work are strictly positive directed so we recommend the reader to pay attention to this fact.

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Liu, Xinjin, and Xuzhong Su. "Theoretical study of Solospun yarn torque." International Journal of Clothing Science and Technology 27, no. 5 (September 7, 2015): 628–39. http://dx.doi.org/10.1108/ijcst-04-2014-0050.

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Purpose – Solospun technology is one of the most important new spinning methods, which is implemented by dividing Ring spinning triangle into several small primary triangles and one final triangle by a Solospun roller. That is, there are two parts of spinning triangle in the Solospun technology, including primary triangles part and final triangle part. In the general case, the primary triangles are much smaller than final triangle. Therefore, the purpose of this paper is to present theoretical study of Solospun yarn torqueby linking the fiber tension in the spinning triangle to yarn torque under the assumption that the primary triangles and the primary twist are ignored. Design/methodology/approach – The theoretical model of the residual torque within Solospun yarn due to the fiber tension was given. Then, as an application of the proposed method, 14.6 tex cotton yarns were taken as an example for the numerical simulations. The fiber tension in the Solospun spinning triangles and corresponding yarn torque were simulated numerically by using Matlab software. The relationships between yarn torque and spinning triangle parameters are analyzed according to the simulation results. Furthermore, the properties of spun yarns produced by the Solospun and Ring spinning system are evaluated and analyzed by using the simulation results. Findings – It is shown that comparing with the Ring spun yarn, Solospun yarn torque is a little larger. Meanwhile, with an increase of substrand number, the fluctuation of curve of average fiber tension in Solospun system is increased firstly, and then decreased, i.e. showing parabola shape, potentially leading to corresponding change of yarn torque. Originality/value – In this paper, theoretical study of Solospun yarn torque is presented by linking the fiber tension in the spinning triangle to yarn torque under the assumption that the primary triangles and the primary twist are ignored. The theoretical model of the residual torque within Solospun yarn due to the fiber tension was given.

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26

YUSTER, RAPHAEL. "Dense Graphs With a Large Triangle Cover Have a Large Triangle Packing." Combinatorics, Probability and Computing 21, no. 6 (September 27, 2012): 952–62. http://dx.doi.org/10.1017/s0963548312000235.

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It is well known that a graph with m edges can be made triangle-free by removing (slightly less than) m/2 edges. On the other hand, there are many classes of graphs which are hard to make triangle-free, in the sense that it is necessary to remove roughly m/2 edges in order to eliminate all triangles.We prove that dense graphs that are hard to make triangle-free have a large packing of pairwise edge-disjoint triangles. In particular, they have more than m(1/4+cβ) pairwise edge-disjoint triangles where β is the density of the graph and c ≥ is an absolute constant. This improves upon a previous m(1/4−o(1)) bound which follows from the asymptotic validity of Tuza's conjecture for dense graphs. We conjecture that such graphs have an asymptotically optimal triangle packing of size m(1/3−o(1)).We extend our result from triangles to larger cliques and odd cycles.

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Bicua, Levi L., Precious R. Tayaben, and Abraham P. Racca. "An Iterative Process Applied to Equilateral Triangles Resulting to the Identity 1/3+1/9+1/27+…=1/2." Abstract Proceedings International Scholars Conference 6, no. 1 (August 13, 2019): 146. http://dx.doi.org/10.35974/isc.v6i1.1207.

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In this paper, we define a particular iterative process and apply it to triangles. It was shown that the process results to self-similar triangles if and only if the generator triangle is an equilateral triangle. Furthermore, the identity 1/3+1/9+1/27+...=1/2 resulted from the process.

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Read, Emrys. "Integer triangles with integer circumradii." Mathematical Gazette 107, no. 569 (July 2023): 241–48. http://dx.doi.org/10.1017/mag.2023.55.

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When studying properties of the circumcircle of an integer triangle, it quickly becomes evident that the radius of such a circle (circumradius) need not itself be an integer. When it is not an integer, the circumradius can still be rational but it can also be irrational, as exemplified in the following examples. It is left to the reader to verify that the triangle with sides 10, 24, 26 has circumradius 13, and that the corresponding values for the triangles with sides 13, 14, 15 and 1, 1, 1 are 65/8 and respectively. It is shown in Theorem 1 below that a necessary condition for the circumradius to be an integer is that the area of the triangle is itself an integer (Heronian triangle) but this condition is not in itself sufficient. A simple counterexample is given by the 13, 14, 15 triangle above which has area 84. However, as a consequence of Theorem 1, we can restrict ourselves to considering Heronian triangles, and relevant properties of such triangles, proved in [1], are given in Theorems 2 and 3 below. We also need to quote some well-known results involving the sums of two squares (see e.g. [2]) and these are listed in Lemma 3. In all that follows, we will use the convention that if T is a triangle with sides a, b, c and z > 0, then zT will denote the triangle, similar to T, with sides za, zb, zc.

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Hansen, David W. "On Inscribed and Escribed Circles of Right Triangles, Circumscribed Triangles, and the Four-Square, Three-Square Problem." Mathematics Teacher 96, no. 5 (May 2003): 358–64. http://dx.doi.org/10.5951/mt.96.5.0358.

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For more than twenty years, I have been studying the fascinating relationships between right triangles and their inscribed and escribed circles. An inscribed circle is one that is tangent to all three sides of a triangle and whose center lies inside the triangle. An escribed circle is one that is tangent to one of the sides of the triangle and to the extensions of the other two sides and whose center lies outside the triangle. Every triangle has one inscribed and three escribed circles, as shown in figure 1.

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Mannion, David. "A Markov chain of triangle shapes." Advances in Applied Probability 20, no. 2 (June 1988): 348–70. http://dx.doi.org/10.2307/1427394.

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The process of choosing a random triangle inside a compact convex region, K, may be iterated when K itself is a triangle. In this way successive generations of random triangles are created. Properties of scale, location and orientation are filtered out, leaving only the shapes of the triangles as the objects of study. Various simulation investigations indicate quite clearly that, as n increases, the nth-generation triangle shape converges to collinearity. In this paper we attempt to establish such convergence; our results fall slightly short of a complete proof.

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Mannion, David. "A Markov chain of triangle shapes." Advances in Applied Probability 20, no. 02 (June 1988): 348–70. http://dx.doi.org/10.1017/s0001867800017018.

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The process of choosing a random triangle inside a compact convex region, K, may be iterated when K itself is a triangle. In this way successive generations of random triangles are created. Properties of scale, location and orientation are filtered out, leaving only the shapes of the triangles as the objects of study. Various simulation investigations indicate quite clearly that, as n increases, the nth-generation triangle shape converges to collinearity. In this paper we attempt to establish such convergence; our results fall slightly short of a complete proof.

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McKay, Angela. "An Analogue of Napoleon’s Theorem in the Hyperbolic Plane." Canadian Mathematical Bulletin 44, no. 3 (September 1, 2001): 292–97. http://dx.doi.org/10.4153/cmb-2001-029-3.

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AbstractThere is a theorem, usually attributed to Napoleon, which states that if one takes any triangle in the Euclidean Plane, constructs equilateral triangles on each of its sides, and connects the midpoints of the three equilateral triangles, one will obtain an equilateral triangle. We consider an analogue of this problem in the hyperbolic plane.

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Ferreira, Luis Dias. "Arithmetic Triangle." Journal of Mathematics Research 9, no. 2 (March 21, 2017): 100. http://dx.doi.org/10.5539/jmr.v9n2p100.

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The product of the first $n$ terms of an arithmetic progression may be developed in a polynomial of $n$ terms. Each one of them presents a coefficient $C_{nk}$ that is independent from the initial term and the common difference of the progression. The most interesting point is that one may construct an "Arithmetic Triangle'', displaying these coefficients, in a similar way one does with Pascal's Triangle. Moreover, some remarkable properties, mainly concerning factorials, characterize the Triangle. Other related `triangles' -- eventually treated as matrices -- also display curious facts, in their linear \emph{modus operandi}, such as successive "descendances''.

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Liu, Hongmin, Hongya Zhang, Zhiheng Wang, and Yiming Zheng. "Feature Matching Based on Triangle Guidance and Constraints." International Journal of Pattern Recognition and Artificial Intelligence 32, no. 08 (April 8, 2018): 1855014. http://dx.doi.org/10.1142/s0218001418550145.

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For images with distortions or repetitive patterns, the existing matching methods usually work well just on one of the two kinds of images. In this paper, we present novel triangle guidance and constraints (TGC)-based feature matching method, which can achieve good results on both kinds of images. We first extract stable matched feature points and combine these points into triangles as the initial matched triangles, and triangles combined by feature points are as the candidates to be matched. Then, triangle guidance based on the connection relationship via the shared feature point between the matched triangles and the candidates is defined to find the potential matching triangles. Triangle constraints, specially the location of a vertex relative to the inscribed circle center of the triangle, the scale represented by the ratio of corresponding side lengths of two matching triangles and the included angles between the sides of two triangles with connection relationship, are subsequently used to verify the potential matches and obtain the correct ones. Comparative experiments show that the proposed TGC can increase the number of the matched points with high accuracy under various image transformations, especially more effective on images with distortions or repetitive patterns due to the fact that the triangular structure are not only stable to image transformations but also provides more geometric constraints.

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Khabibah, Siti. "PEWARNAAN PADA GRAF BINTANG SIERPINSKI." Jurnal Ilmiah Matematika dan Pendidikan Matematika 9, no. 1 (June 23, 2017): 37. http://dx.doi.org/10.20884/1.jmp.2017.9.1.2853.

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This paper discuss about Sierpinski star graph , which its construction based on the Sierpinski triangle. Vertex set of Sierpinski star graph is a set of all triangles in Sierpinski triangle; and the edge set of Sierpinski star graph is a set of all sides that are joint edges of two triangles on Sierpinski triangle. From the vertex and edge coloring of Sierpinski star graph, it is found that the chromatic number on vertex coloring of graph is 1 for n = 1 and 2 for ; while the chromatic number on edge coloring of graf is 0 for n = 1 and for

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CALVO, T., G. MAYOR, J. TORRENS, J. SUÑER, M. MAS, and M. CARBONELL. "GENERATION OF WEIGHTING TRIANGLES ASSOCIATED WITH AGGREGATION FUNCTIONS." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 08, no. 04 (August 2000): 417–51. http://dx.doi.org/10.1142/s0218488500000290.

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In this work, we present several ways to obtain different types of weighting triangles, due to these types characterize some interesting properties of Extended Ordered Weighted Averaging operators, EOWA, and Extended Quasi-linear Weighted Mean, EQLWM, as well as of their reverse functions. We show that any quantifier determines an EOWA operator which is also an Extended Aggregation Function, EAF, i.e., the weighting triangle generated by a quantifier is always regular. Moreover, we present different results about generation of weighting triangles by means of sequences and fractal structures. Finally, we introduce a degree of orness of a weighting triangle associated with an EOWA operator. After that, we mention some results on each class of triangle, considering each one of these triangles as triangles associated with their corresponding EOWA operator, and we calculate the ornessof some interesting examples.

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37

Crilly, Tony, and Colin R. Fletcher. "Triangles meeting triangles." Mathematical Gazette 98, no. 543 (November 2014): 432–51. http://dx.doi.org/10.1017/s0025557200008135.

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We consider two connected problems: •For a given but otherwise arbitrary triangle in the plane, to construct similar triangles which ‘meet’ this triangle.•To find the triangle so formed which has least area.1. Constructing a triangle which meets anotherThese problems beg the question of what is meant by ‘meet’ and we now aim to make this precise: Definition: A triangle XYZ will meet a given triangle ABC if on the triangle ABC, the vertex X lies on a line through AB, the vertex Y lies on a line through BC, and the vertex Z lies on a line through CA.When triangle XYZ is actually ‘in’ the triangle ABC, ‘meet’ is synonymous with the traditional ‘inscribe’ (such as in case (1) below). For ‘inscribe’ we understand that some of X, Y, Z may coincide with the vertices of ABC (such as case (2) below).More generally we use ‘meet’ to extend these possibilities by allowing XYZ to meet triangle ABC with its sides produced externally (such as case (3) below).

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Du, Jian Jun, Xin Yu Guo, Sheng Lian Lu, Bo Xiang Xiao, and Jian Wei Wu. "Building Three-Dimensional Merged Surface Model from Polygonal Models." Advanced Materials Research 566 (September 2012): 336–41. http://dx.doi.org/10.4028/www.scientific.net/amr.566.336.

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Three-dimensional surface merging plays an important role in rapid prototyping manufacture, physically based modeling and finite element analysis. In this paper, a rapid merging method is proposed to build three-dimensional water-proof surface model from polygonal models. To rapidly determine merging boundaries, collision detection techniques are used to obtain the intersection triangle pairs between the two input models, and then the intersection line loops are accurately computed. Furthermore, triangle tessellation and edge searching method is used to generate new triangles and classify each triangle in models into different triangle sets. Finally, an inclusion test determines the position of each triangle set and stitches the labeled triangle sets into the merged model. The experimental results demonstrate the robustness and adaptability of the presented method.

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39

Silvester, John R. "On cardioids and Morley's theorem." Mathematical Gazette 105, no. 562 (February 17, 2021): 40–51. http://dx.doi.org/10.1017/mag.2021.6.

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Morley’s trisector theorem says that the three intersections of the trisectors of the angles of a triangle, lying near the three sides respectively, form an equilateral triangle. See Figure 1. Morley discovered his theorem in 1899, and news of it quickly spread. Over the years many proofs have been published, by trigonometry or by geometry, but a simple angle-chasing argument is elusive. See [1] for a list up to 1978. Perhaps the easiest proof is that of John Conway [2], who assembles a triangle similar to the given triangle by starting with an equilateral triangle and surrounding it by triangles with very carefully chosen angles.

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Deng, Tian-Bo. "Generalized Stability-Triangle for Guaranteeing the Stability-Margin of the Second-Order Digital Filter." Journal of Circuits, Systems and Computers 25, no. 08 (May 17, 2016): 1650094. http://dx.doi.org/10.1142/s0218126616500948.

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In the design of recursive digital filters, the stability of the recursive digital filters must be guaranteed. Furthermore, it is desirable to add a certain amount of margin to the stability so as to avoid the violation of stability due to some uncertain perturbations of the filter coefficients. This paper extends the well-known stability-triangle of the second-order digital filter into more general cases, which results in dented stability-triangles and generalized stability-triangle. The generalized stability-triangle can be viewed as a special case of the dented stability-triangles if the two upper bounds on the radii of the two poles are the same, which is a generalized version of the existing conventional stability-triangle and can guarantee the radii of the two poles of the second-order recursive digital filter below some prescribed upper bound. That is, it is able to provide a prescribed stability-margin in terms of the upper bound of the pole radii. As a result, the generalized stability-triangle increases the flexibility for guaranteeing a prescribed stability-margin. Since the generalized stability-triangle is parameterized by using the upper bound of pole radii, i.e., the stability-margin is parameterized as a function of the upper bound, the proposed generalized stability-triangle facilitates the stability-margin guarantee in the design of the second-order as well as high-order recursive digital filters.

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Padrón, Miguel Ángel, Francisco Perdomo, Ángel Plaza, and José Pablo Suárez. "The Shortest-Edge Duplication of Triangles." Mathematics 10, no. 19 (October 5, 2022): 3643. http://dx.doi.org/10.3390/math10193643.

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We introduce a new triangle transformation, the shortest-edge (SE) duplication, as a natural way of mesh derefinement suitable to those meshes obtained by iterative application of longest-edge bisection refinement. Metric properties of the SE duplication of a triangle in the region of normalised triangles endowed with the Poincare hyperbolic metric are studied. The self-improvement of this transformation is easily proven, as well as the minimum angle condition. We give a lower bound for the maximum of the smallest angles of the triangles produced by the iterative SE duplication α=π6. This bound does not depend on the shape of the initial triangle.

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Neggazi, Brahim, Volker Turau, Mohammed Haddad, and Hamamache Kheddouci. "A O(m) Self-Stabilizing Algorithm for Maximal Triangle Partition of General Graphs." Parallel Processing Letters 27, no. 02 (June 2017): 1750004. http://dx.doi.org/10.1142/s0129626417500049.

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The triangle partition problem is a generalization of the well-known graph matching problem consisting of finding the maximum number of independent edges in a given graph, i.e., edges with no common node. Triangle partition instead aims to find the maximum number of disjoint triangles. The triangle partition problem is known to be NP-complete. Thus, in this paper, the focus is on the local maximization variant, called maximal triangle partition (MTP). Thus, paper presents a new self-stabilizing algorithm for MTP that converges in O(m) moves under the unfair distributed daemon.

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Jørgensen, Peter. "Auslander-Reiten triangles in subcategories." Journal of K-theory 3, no. 3 (November 14, 2008): 583–601. http://dx.doi.org/10.1017/is008007021jkt056.

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AbstractThis paper studies Auslander-Reiten triangles in subcategories of triangulated categories. The main theorem shows that the Auslander-Reiten triangles in a subcategory are closely connected with the approximation properties of the subcategory. Namely, let C be an object in the subcategory C of the triangulated category T, and letbe an Auslander-Reiten triangle in T. Then under suitable assumptions, there is an Auslander-Reiten trianglein C if and only if there is a minimal right-C-approximation of the form.The theory is used to give a new proof of the existence of Auslander-Reiten sequences over finite dimensional algebras.

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Pambuccian, Victor. "Acute Triangulation of a Triangle in a General Setting." Canadian Mathematical Bulletin 53, no. 3 (September 1, 2010): 534–41. http://dx.doi.org/10.4153/cmb-2010-059-4.

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AbstractWe prove that, in ordered plane geometries endowed with a very weak notion of orthogonality, one can always triangulate any triangle into seven acute triangles, and, in case the given triangle is not acute, into no fewer than seven.

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Liu, Xinjin, and Xuzhong Su. "Theoretical Study of Effect of Ring Spinning Triangle Division on Fiber Tension Distribution." Journal of Engineered Fibers and Fabrics 10, no. 3 (September 2015): 155892501501000. http://dx.doi.org/10.1177/155892501501000314.

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The spinning triangle is a critical region in the spinning process of staple yarn. Its geometry influences the distribution of fiber tension at spinning triangle directly and affects the qualities of spun yarns. Taking appropriate measures to change the ring spinning triangle geometry and improve the qualities of yarn has attracted more and more interest recently. Spinning triangle division is one of the most effective measures, such as solospun technology. Therefore, in this paper, the effect of ring spinning triangle division on fiber tension distribution was studied theoretically. The general case that the spinning triangle is divided into two parts, primary triangle and final triangle, was investigated. Firstly, theoretical models of the fiber tension distributions in the final and primary spinning triangles are given respectively using the principle of minimum potential energy. Secondly, the fiber tension distributions in the spinning triangle with different shape parameters are numerically simulated. Finally, the possible effects of ring spinning triangle division on yarn qualities are analyzed according to the numerical simulations and previous results. In addition, the properties of spun yarns produced by the modified ring spinning system were evaluated and analyzed.

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Hwang, Seokhoon, Junehee Lee, Yongsun Kim, Hyeonseok Na, and Youngmin Cho. "A Study on the Exellipse of Triangle." Korean Science Education Society for the Gifted 14, no. 3 (December 30, 2022): 218–28. http://dx.doi.org/10.29306/jseg.2022.14.3.218.

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This study was based on the research results conducted as a YSC project. Studies on the inellipse of a triangle and previous studies that explored the properties between the incircle and the excircle of a triangle were analyzed. I wondered if the excircle of a triangle could be extended to an exellipse, and whether the properties that exist between an incircle and a excircle of a triangle also hold true between an inellipse and an exellipse of a triangle. Therefore, in this study, I defined the exellipse of a triangle and explored the properties. Through this study, the following research results were obtained. First, the exellipse of the triangle was defined, and its existence and uniqueness were proved. Second, we found the division ratio at which the exellipse internally and externally divides the line segment and extension line of a triangle. Third, it was revealed that various properties including the Lurier theorem for ellipses were established in triangles. Fourth, a method of constructing an exellipse of a triangle was discovered. Based on this study, it is expected that follow-up studies on the exellipse of the triangle and the expansion of the various triangle centers will be actively conducted through this study.

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Yuan, Liping, and Ren Ding. "Triangles in squares." Mathematical Gazette 88, no. 512 (July 2004): 219–25. http://dx.doi.org/10.1017/s0025557200174947.

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In this paper we find the necessary and sufficient conditions on a, b, c, s for a triangle with sides a, b, c to fit into a square of side s.Questions about precisely when one shape fits into another attract wide attention. In 1993 Post [1] gave necessary and sufficient conditions on the six sides of two triangles for the first to fit into the second. Recently, the necessary and sufficient conditions for squares to fit in triangles [2], equilateral triangles in triangles [3], rectangles in triangles [4] and rectangles in rectangles [5] are given. In [2] Wetzel asked when a given triangle fits into a given square. In this paper we find the necessary and sufficient conditions on a, b, c, s for a triangle with sides a, b, c to fit into a square of side s. For the sake of convenience, let α, β, γ denote the angles opposite sides a, b, c respectively, and we may assume without loss of generality that a ≥ b ≥ c, which implies that α ≥ β ≥ γ.

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Wulandari, Hilaria Yesieka Ayu, and Tatag Yuli Eko Siswono. "Students Activities in Learning Pythagoras Theorems Using Desmos Application." Journal of Mathematical Pedagogy (JoMP) 5, no. 1 (March 1, 2024): 1–14. http://dx.doi.org/10.26740/jomp.v5n1.p1-14.

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This study aims to describe the creative thinking skills of students in solving problems related to the Pythagorean Theorem using Desmos application. This descriptive research involved 3 Junior High School students in grade 9 in Sidoarjo (Indonesia) who each had high, medium, and low ability backgrounds. The task in the form of questions consisted of two questions asking students to create two different triangles. The results of the analysis showed that high-ability students were able to create two different triangles if they were given sides 3 and 4 in the form of a right triangle and an equilateral triangle. Students with moderate and low abilities were able to draw right triangles with Desmos, but had difficulty drawing isosceles triangles. Both of them drew an isosceles triangle, but with side lengths of 3, 4, and 6. These results illustrate that the use of Desmos helps students' ideas in finding answers in drawing triangles, but the basic abilities that students have can be an obstacle in completing the given tasks.

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Adams, Thomasenia Lott, and Fatma Aslan-Tutak. "Math Roots: Serving Up Sierpinski!" Mathematics Teaching in the Middle School 11, no. 5 (January 2006): 248–53. http://dx.doi.org/10.5951/mtms.11.5.0248.

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The sierpinski triangle, created in 1916, has some very interesting characteristics. It is an impressive and valuable topic for mathematical exploration, since it combines Euclidean geometry (triangles and measurement) with fractal geometry. The Sierpinski triangle, also known as the Sierpinski gasket, is a fractal formed from an equilateral triangle. It is one of the most popular fractals to construct and analyze in middle school mathematics lessons. Since the 1960s, it has been possible to design fractals using a computer program, especially the complex examples that are often difficult to construct by hand. However, students can easily duplicate the Sierpinski triangle.

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Andrica, Dorin, and Ovidiu Bagdasar. "On Some Properties of the Equilateral Triangles with Vertices Located on the Support Sides of a Triangle." Axioms 13, no. 7 (July 17, 2024): 478. http://dx.doi.org/10.3390/axioms13070478.

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The possible positions of an equilateral triangle whose vertices are located on the support sides of a generic triangle are studied. Using complex coordinates, we show that there are infinitely many such configurations, then we prove that the centroids of these equilateral triangles are collinear, defining two lines perpendicular to the Euler’s line of the original triangle. Finally, we obtain the complex coordinates of the intersection points and study some particular cases.

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