Journal articles: 'Similar Triangles'

1

Zak, Andrzej. "Dissection of a Triangle into Similar Triangles." Discrete & Computational Geometry 34, no. 2 (March 18, 2005): 295–312. http://dx.doi.org/10.1007/s00454-005-1167-1.

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Biber, Abdullah Çağrı. "Students' difficulties in similar triangle questions." Cypriot Journal of Educational Sciences 15, no. 5 (October 29, 2020): 1146–59. http://dx.doi.org/10.18844/cjes.v15i5.5161.

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Similar triangles in questions are usually given as separate, adjacent or overlapped. Furthermore, similarity types such as Side-Angle-Side (S.A.S.), Side-Side-Side (S.S.S.) and Angle-Angle (A.A.) are requested in the questions. Students have more trouble in other types of questions. The purpose of this study is to investigate the difficulties of students about similar triangles and the reasons for these difficulties. This research was carried out with the case study method, which is one of the qualitative research approaches. The study was conducted with 55 Science High School 9th grade students and 9 open-ended questions were used to examine students' knowledge about “similarity in triangles”. Furthermore, 5 students were interviewed to find out the reasons for their solutions. Descriptive analysis method was used to analyze the data. As a result, it can be concluded that students have difficulties mostly in overlapped triangles and Angle-Angle type questions. On the other hand, it can be concluded that students are quite successful where similar triangles are given separately. In the light of the findings obtained in this study, it can be advised for lecturers to focus on the questions where similar triangles are overlapped while explaining the similarity in the triangle. Keywords: Similarity, Triangles, Difficulties, High School Students.

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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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Jones, C. A., P. Jones, and A. B. Bolt. "Dissections of Triangles into Five Similar Triangles." Mathematical Gazette 82, no. 494 (July 1998): 225. http://dx.doi.org/10.2307/3620405.

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5

Okumura, Hiroshi. "79.56 Two Similar Triangles." Mathematical Gazette 79, no. 486 (November 1995): 569. http://dx.doi.org/10.2307/3618096.

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6

Johnson, Gwen. "Mathematical Exploration: Similar Triangles." Mathematics Teaching in the Middle School 16, no. 4 (November 2010): 248–53. http://dx.doi.org/10.5951/mtms.16.4.0248.

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Within school mathematics, similarity is a topic that has many connections to real life. Similarity is related to scale drawings, such as those used in architecture, and models, including models of trains, cars, and boats. It is also used to solve problems that involve similar figures, which are common in middle-grades textbooks.

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Richardson, Thomas J. "Optimal packing of similar triangles." Journal of Combinatorial Theory, Series A 69, no. 2 (February 1995): 288–300. http://dx.doi.org/10.1016/0097-3165(95)90054-3.

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8

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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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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10

Ramachandra, K. "86.54 Pythagoras' Theorem and Similar Triangles." Mathematical Gazette 86, no. 506 (July 2002): 324. http://dx.doi.org/10.2307/3621877.

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11

Leung, King-Shun. "Similar triangles and the Cosine Rule." Mathematical Gazette 96, no. 535 (March 2012): 169–72. http://dx.doi.org/10.1017/s0025557200004290.

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Klinaku, Shukri, and Valbone Berisha. "The Doppler effect and similar triangles." Results in Physics 12 (March 2019): 846–52. http://dx.doi.org/10.1016/j.rinp.2018.12.024.

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13

Su, Zhanjun, and Ren Ding. "Tilings of parallelograms with similar triangles." Journal of Applied Mathematics and Computing 23, no. 1-2 (January 2007): 321–27. http://dx.doi.org/10.1007/bf02831978.

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14

Laczkovich, M. "Tilings of polygons with similar triangles." Combinatorica 10, no. 3 (September 1990): 281–306. http://dx.doi.org/10.1007/bf02122782.

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15

Matsuura, Ryota. "Delving Deeper: Approximating π Using Similar Triangles." Mathematics Teacher 105, no. 8 (April 2012): 632–36. http://dx.doi.org/10.5951/mathteacher.105.8.0632.

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This article presents a method for approximating π using similar triangles that was inspired by the author's work with middle school teachers. The method relies on a repeated application of a geometric construction that allows us to inscribe regular polygons inside a unit circle with arbitrarily large number of sides.

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Su, Zhanjun, Chan Yin, Xiaobing Ma, and Ying Li. "Tilings of Parallelograms with Similar Right Triangles." Discrete & Computational Geometry 50, no. 2 (June 18, 2013): 469–73. http://dx.doi.org/10.1007/s00454-013-9522-0.

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17

QUIGGIN, JOHN. "SIMILAR TRIANGLES AND THE THEORY OF REGULATION." Economic Papers: A journal of applied economics and policy 14, no. 3 (September 1995): 86–88. http://dx.doi.org/10.1111/j.1759-3441.1995.tb00915.x.

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18

Wang, David G. L. "Tilings of Parallelograms by Similar Isosceles Triangles." Mathematical Intelligencer 38, no. 3 (June 6, 2016): 24–29. http://dx.doi.org/10.1007/s00283-016-9629-2.

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19

Laczkovich, M. "Tilings of Polygons with Similar Triangles, II." Discrete & Computational Geometry 19, no. 3 (March 1998): 411–25. http://dx.doi.org/10.1007/pl00009359.

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20

Scott, J. A., and H. M. Cundy. "85.64 Similar Triangles Associated with the Lemoine Point." Mathematical Gazette 85, no. 504 (November 2001): 486. http://dx.doi.org/10.2307/3621766.

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21

Su, Zhanjun, and Lingnan He. "Tilings of regular pentagons with similar right triangles." Rocky Mountain Journal of Mathematics 50, no. 5 (October 2020): 1853–57. http://dx.doi.org/10.1216/rmj.2020.50.1853.

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22

Szegedy, Balázs. "Tilings of the Square With Similar Right Triangles." Combinatorica 21, no. 1 (January 1, 2001): 139–44. http://dx.doi.org/10.1007/s004930170008.

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23

Fox, Michael. "Some properties of Kiepert lines of a triangle." Mathematical Gazette 100, no. 547 (March 2016): 54–67. http://dx.doi.org/10.1017/mag.2016.8.

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This article describes an investigation into Kiepert lines, and leads to some surprising and little-known relationships between the Fermat, Napoleon and Vecten points of a triangle.If we draw similar isosceles triangles A'BC, B'CA and C'AB outwards on the sides of a given scalene triangle ABC as in Figure 1, Kiepert's theorem tells us that the lines A'A, B'B and C'C meet in a single point - a Kiepert point [1, Chapter 11]. Since its position depends on the common base angles θ of the isosceles triangles, I label it K(θ), taking θ as the parameter of this point.

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24

Cuicchi, Paul M., and Paul S. Hutchison. "Using a Simple Optical Rangefinder to Teach Similar Triangles." Mathematics Teacher 96, no. 3 (March 2003): 166–68. http://dx.doi.org/10.5951/mt.96.3.0166.

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Simple range-finding and height-finding devices that use the properties of similar triangles are common. For example, foresters use these devices to measure the heights of trees, and some golfers use them to measure the distance from ball to hole, although such devices are illegal in competitive golf. Simple optical range-finding golf scopes are inexpensive and are reasonably accurate.

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Su, Zhanjun, and Ren Ding. "Tilings of orthogonal polygons with similar rectangles or triangles." Journal of Applied Mathematics and Computing 17, no. 1-2 (March 2005): 343–50. http://dx.doi.org/10.1007/bf02936060.

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26

Leversha, Gerry, and G. C. Smith. "Euler and triangle geometry." Mathematical Gazette 91, no. 522 (November 2007): 436–52. http://dx.doi.org/10.1017/s0025557200182087.

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There is a very easy way to produce the Euler line, using transformational arguments. Given a triangle ABC, let AʹBʹ'C be the medial triangle, whose vertices are the midpoints of the sides. These two triangles are homothetic: they are similar and corresponding sides are parallel, and the centroid, G, is their centre of similitude. Alternatively, we say that AʹBʹC can be mapped to ABC by means of an enlargement, centre G, with scale factor –2.

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27

Guo, C. Q., Y. Yan, L. Li, J. X. Han, K. Wei, F. Lin, and H. Zhang. "Stable local structure in LSMCO system and its relationship with pseudogap." International Journal of Modern Physics B 29, no. 25n26 (October 14, 2015): 1542028. http://dx.doi.org/10.1142/s021797921542028x.

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Pseudogap is a unique property of high-[Formula: see text] superconductors. However, it was also found that there is a similar pseudogap in giant magnetoresistive La–Sr–Mn–O materials, which are not superconductors. In our previous research, we suggested that the origin of the pseudogap is correlated with the stability of [Formula: see text] plane. We call it a steady “fixed triangle” local structure. If our assumption is correct, it will be expected to find a similar stable local structure in the La–Sr–Mn–O system, which is similar to the [Formula: see text] plane in cuprates. We synthesized Co-doped [Formula: see text] (LSMCO) materials and investigated them. Based on the detailed analysis of the crystalline structure, it is found that there indeed exists a stable local triangles in [Formula: see text] plane. Comparing this similarity with the fixed triangles in the cuprate superconductors, it shows that the origin of the pseudogap may possibly be correlated with the stability of the local structure.

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28

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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29

Spirkovska, Lilly. "Three-dimensional object recognition using similar triangles and decision trees." Pattern Recognition 26, no. 5 (May 1993): 727–32. http://dx.doi.org/10.1016/0031-3203(93)90125-g.

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30

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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31

Sridevi, P. "A Note on Detection of Communities in Social Networks." International Journal of Engineering and Computer Science 9, no. 03 (March 19, 2020): 24978–83. http://dx.doi.org/10.18535/ijecs/v9i03.4452.

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The modern Science of Social Networks has brought significant advances to our understanding of the Structure, dynamics and evolution of the Network. One of the important features of graphs representing the Social Networks is community structure. The communities can be considered as fairly independent components of the social graph that helps identify groups of users with similar interests, locations, friends, or occupations. The community structure is closely tied to triangles and their count forms the basis of community detection algorithms. The present work takes into consideration, a triangle instead of the edge as the basic indicator of a strong relation in the social graph. A simple triangle counting algorithm is then used to evaluate different metrics that are employed to detect strong communities. The methodology is applied to Zachary Social network and discussed. The results bring out the usefulness of counting triangles in a network to detect strong communities in a Social Network.

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32

ŻAK, ANDRZEJ. "DISSECTIONS OF POLYGONS INTO CONVEX POLYGONS." International Journal of Computational Geometry & Applications 20, no. 02 (April 2010): 223–44. http://dx.doi.org/10.1142/s021819591000327x.

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In this paper we present purely combinatorial conditions that allow us to recognize the topological equivalence (or non-equivalence) of two given dissections. Using a computer program based on this result, we are able to generate a set which contains all topologically non-equivalent dissections of a p0-gon into convex pi-gons, i = 1…n, where n, p0,…,pn are integers such that n ≥ 2, pi ≥ 3. By analyzing generated structures, we are able to find all (up to similarity) dissections of a given type. Since the number of topologically non-equivalent dissections is huge even when the number of parts is small, it is necessary to find additional combinatorial conditions depending on the type of sought dissections, which will allow us to exclude the majority of generated structures. We present such conditions for some special dissections of a triangle into triangles. Finally we prove two new results concerning perfect dissections of a triangle into similar triangles.

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33

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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34

Hirshfeld, Alan W. "The Triangles of Aristarchus." Mathematics Teacher 97, no. 4 (April 2004): 228–31. http://dx.doi.org/10.5951/mt.97.4.0228.

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The ancient Greek mathematician Aristarchus demonstrated for the first time how it was possible, using simple observations and elementary geometry, to measure distances to bodies in the solar system. Aristarchus' methods used a lunar elcipse to approximate the diameter of the Earth, and used the shadow cone of a lunar eclipse to form similar triangles and proportional measurements. The mathmatics can be easily understood by a high school geometry student.

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35

Guo, Kehua, Yongling Liu, and Guihua Duan. "Differential and Statistical Approach to Partial Model Matching." Mathematical Problems in Engineering 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/249847.

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Partial model matching approaches are important to target recognition. In this paper, aiming at a 3D model, a novel solution utilizing Gaussian curvature and mean curvature to represent the inherent structure of a spatial shape is proposed. Firstly, a Point-Pair Set is constructed by means of filtrating points with a similar inherent characteristic in the partial surface. Secondly, a Triangle-Pair Set is demonstrated after locating the spatial model by asymmetry triangle skeleton. Finally, after searching similar triangles in a Point-Pair Set, optimal transformation is obtained by computing the scoring function in a Triangle-Pair Set, and optimal matching is determined. Experiments show that this algorithm is suitable for partial model matching. Encouraging matching efficiency, speed, and running time complexity to irregular models are indicated in the study.

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36

DOLLERY, BRIAN. "SIMILAR TRIANGLES AND THE THEORY OF REGULATION: A REJOINDER TO QUIGGIN." Economic Papers: A journal of applied economics and policy 15, no. 1 (March 1996): 94–96. http://dx.doi.org/10.1111/j.1759-3441.1996.tb00928.x.

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37

Sedaghat, Hassan. "Measuring the Speed and Altitude of an Aircraft Using Similar Triangles." SIAM Review 33, no. 4 (December 1991): 650–54. http://dx.doi.org/10.1137/1033139.

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Fujiwara, Toshiaki, Hiroshi Fukuda, Atsushi Kameyama, Hiroshi Ozaki, and Michio Yamada. "Synchronized similar triangles for three-body orbits with zero angular momentum." Journal of Physics A: Mathematical and General 37, no. 44 (October 21, 2004): 10571–84. http://dx.doi.org/10.1088/0305-4470/37/44/008.

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SHPARLINSKI, IGOR E. "ON POINT SETS IN VECTOR SPACES OVER FINITE FIELDS THAT DETERMINE ONLY ACUTE ANGLE TRIANGLES." Bulletin of the Australian Mathematical Society 81, no. 1 (October 21, 2009): 114–20. http://dx.doi.org/10.1017/s0004972709000719.

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AbstractFor three points$\vec {u}$,$\vec {v}$and$\vec {w}$in then-dimensional space 𝔽nqover the finite field 𝔽qofqelements we give a natural interpretation of an acute angle triangle defined by these points. We obtain an upper bound on the size of a set 𝒵 such that all triples of distinct points$\vec {u}, \vec {v}, \vec {w} \in \cZ $define acute angle triangles. A similar question in the real space ℛndates back to P. Erdős and has been studied by several authors.

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Gabrielli, Andrea, and Mario V. Wüthrich. "Back-testing the chain-ladder method." Annals of Actuarial Science 13, no. 2 (November 13, 2018): 334–59. http://dx.doi.org/10.1017/s1748499518000325.

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AbstractThe chain-ladder method is one of the most popular claims reserving techniques. The aim of this study is to back-test the chain-ladder method. For this purpose, we use a stochastic scenario generator that allows us to simulate arbitrarily many upper claims reserving triangles of similar characteristics for which we also know the corresponding lower triangles. Based on these simulated triangles, we analyse the performance of the chain-ladder claims reserving method.

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Esterman, Michael, Regina Mcglinchey-Berroth, Michael P. Alexander, and William Milberg. "Triangular backgrounds shift line bisection performance in hemispatial neglect: The critical point." Journal of the International Neuropsychological Society 8, no. 5 (July 2002): 721–26. http://dx.doi.org/10.1017/s1355617702801369.

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AbstractIsosceles triangular backgrounds influence line bisection performance in normal control participants and patients with hemispatial neglect. When the triangles are oriented asymmetrically with the vertex in 1 visual field, and the base in the other, the perceived midpoint of horizontal lines within the triangle is shifted towards the base, and away from the vertex. The current study examines this illusion further by systematically varying the extent of the triangle presented. With only fragments of the triangle in the background of the line, the vertex is the critical component driving the illusory shift in perceived midpoint. Patients with neglect and controls are equally sensitive to the illusion. Similar geometric illusions that are also intact in neglect, along with these results, suggest that preattentive, implicit visual processing is preserved in neglect and drives these illusions.

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Mazumdar, Arindam, Somnath Bharadwaj, and Debanjan Sarkar. "Quantifying the redshift space distortion of the bispectrum II: induced non-Gaussianity at second-order perturbation." Monthly Notices of the Royal Astronomical Society 498, no. 3 (August 22, 2020): 3975–84. http://dx.doi.org/10.1093/mnras/staa2548.

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ABSTRACT The anisotrpy of the redshift space bispectrum $B^s(\boldsymbol {k_1},\boldsymbol {k_2},\boldsymbol {k_3})$, which contains a wealth of cosmological information, is completely quantified using multipole moments $\bar{B}^m_{\ell }(k_1,\mu ,t)$, where k1, the length of the largest side, and (μ, t), respectively, quantify the size and the shape of the triangle $(\boldsymbol {k_1},\boldsymbol {k_2},\boldsymbol {k_3})$. We present analytical expressions for all the multipoles that are predicted to be non-zero (ℓ ≤ 8, m ≤ 6) at second-order perturbation theory. The multipoles also depend on β1, b1, and γ2, which quantify the linear redshift distortion parameter, linear bias and quadratic bias, respectively. Considering triangles of all possible shapes, we analyse the shape dependence of all of the multipoles holding $k_1=0.2 \, {\rm Mpc}^{-1}, \beta _1=1, b_1=1$, and γ2 = 0 fixed. The monopole $\bar{B}^0_0$, which is positive everywhere, is minimum for equilateral triangles. $\bar{B}_0^0$ increases towards linear triangles, and is maximum for linear triangles close to the squeezed limit. Both $\bar{B}^0_{2}$ and $\bar{B}^0_4$ are similar to $\bar{B}^0_0$, however, the quadrupole $\bar{B}^0_2$ exceeds $\bar{B}^0_0$ over a significant range of shapes. The other multipoles, many of which become negative, have magnitudes smaller than $\bar{B}^0_0$. In most cases, the maxima or minima, or both, occur very close to the squeezed limit. $\mid \bar{B}^m_{\ell } \mid$ is found to decrease rapidly if ℓ or m are increased. The shape dependence shown here is characteristic of non-linear gravitational clustering. Non-linear bias, if present, will lead to a different shape dependence.

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ISHII, Haruyuki, Akinori MURATA, Fumito UWANO, Takato TATSUMI, Yuta UMENAI, Keiki TAKADAMA, Tomohiro HARADA, et al. "SLIM Spacecraft Location Estimation by Crater Matching Based on Similar Triangles and Its Improvement." AEROSPACE TECHNOLOGY JAPAN, THE JAPAN SOCIETY FOR AERONAUTICAL AND SPACE SCIENCES 17 (2018): 69–78. http://dx.doi.org/10.2322/astj.jsass-d-17-00011.

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Boonklurb, Ratinan, and Eakasit Sanguanlorsit. "(1 + n) sequential dissection of a rectangle into m-GONS, m ∈{3,5,6,7,8}." Discrete Mathematics, Algorithms and Applications 08, no. 03 (August 2016): 1650039. http://dx.doi.org/10.1142/s1793830916500397.

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This paper gives algorithms to dissect a rectangle sequentially into polygons that can be reassembled to a rectangle, which is similar to the initial rectangle, and [Formula: see text]-similar equilateral triangles or regular hexagons and also pentagons or heptagons or octagons with symmetry.

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45

Lundberg, Christina. "Can We Use a Mirror to Find Height?" Mathematics Teacher 110, no. 5 (December 2016): 400–402. http://dx.doi.org/10.5951/mathteacher.110.5.0400.

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Bragg, Don C. "An Improved Tree Height Measurement Technique Tested on Mature Southern Pines." Southern Journal of Applied Forestry 32, no. 1 (February 1, 2008): 38–43. http://dx.doi.org/10.1093/sjaf/32.1.38.

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Abstract Virtually all techniques for tree height determination follow one of two principles: similar triangles or the tangent method. Most people apply the latter approach, which uses the tangents of the angles to the top and bottom and a true horizontal distance to the subject tree. However, few adjust this method for ground slope, tree lean, crown shape, and crown configuration, making errors commonplace. Given documented discrepancies exceeding 30% with current methods, a reevaluation of height measurement is in order. The sine method is an alternative that measures a real point in the crown. Hence, it is not subject to the same assumptions as the similar triangle and tangent approaches. In addition, the sine method is insensitive to distance from tree or observer position and can not overestimate tree height. The advantages of the sine approach are shown with mature southern pines from Arkansas.

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47

ZHU, ZHI-YONG. "LIPSCHITZ EQUIVALENCE OF TOTALLY DISCONNECTED GENERAL SIERPINSKI TRIANGLES." Fractals 23, no. 02 (May 28, 2015): 1550013. http://dx.doi.org/10.1142/s0218348x15500139.

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Given an integer n ≥ 2 and an ordered pair (A, B) with A ⊂ {k1α + k2β : k1 + k2 ≤ n - 1 and k1, k2 ∈ ℕ ∪{0}} and B ⊂ {k1α + k2β : 2 ≤ k1 + k2 ≤ n and k1, k2 ∈ ℕ}, where [Formula: see text]. Let T ≔ T(A, B) be unique compact set of ℝ2 satisfying the set equation: T = [(T+A)∪(B-T)]/n. In this paper, we show that such self-similar sets which are totally disconnected are determined to within Lipschitz equivalence by their Hausdorff dimension.

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48

De Villiers, Michael. "From the Fermat points to the De Villiers3 points of a triangle." Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 29, no. 3 (January 13, 2010): 119–29. http://dx.doi.org/10.4102/satnt.v29i3.16.

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The article starts with a problem of ﬁnding a point that minimizes the sum of the distances to the vertices of an acute-angled triangle, a problem originally posed by Fermat in the 1600’s, and apparently ﬁrst solved by the Italian mathematician and scientist Evangelista Torricelli. This point of optimization is therefore usually called the inner Fermat or Fermat-Torricelli point of a triangle. The transformation proof presented in the article was more recently invented in 1929 by the German mathematician J. Hoffman. After reviewing the centroid and medians of a triangle, these are generalized to Ceva’s theorem, which is then used to prove the following generalization of the Fermat-Torricelli point from [3]: “If triangles DBA, ECB and FAC are constructed outwardly (or inwardly) on the sides of any ∆ABC so that ∠DAB =∠CAF , ∠ DBA = ∠ CBE and ∠ ECB = ∠ ACF then DC, EA and FB are concurrent.”However, this generalization is not new, and the earliest proof the author could trace is from 1936 by W. Hoffer in [1], though the presented proof is distinctly different. Of practical relevance is the fact that this Fermat-Torricelli generalization can be used to solve a “weighted” airport problem, for example, when the populations in the three cities are of different size. The author was also contacted via e-mail in July 2008 by Stephen Doro from the College of Physicians and Surgeons, Columbia University, USA, who was considering its possible application in the branching of larger arteries and veins in the human body into smaller and smaller ones. On the basis of an often-observed (but not generally true) duality between circumcentres and in centres, it was conjectured in 1996 [see 4] that the following might be true from a similar result for circumcentres (Kosnita’s theorem), namely: The lines joining the vertices A, B, and C of a given triangle ABC with the incentres of the triangles BCO, CAO, and ABO (O is the incentre of ∆ABC), respectively, are concurrent (in what is now called the inner De Villiers point). Investigation on the dynamic geometry program Sketchpad quickly conﬁ rmed that the conjecture was indeed true. (For an interactive sketch online, see [7]). Using the aforementioned generalization of the Fermat-Torricelli point, it was now also very easy to prove this result. The outer De Villiers point is similarly obtained when the excircles are constructed for a given triangle ABC, in which case the lines joining the vertices A, B, and C of a given triangle ABC with the incentres of the triangles BCI1, CAI2, and ABI3 (Ii are the excentres of ∆ABC), are concurrent. The proof follows similarly from the Fermat-Torricelli generalization.

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Laurinen, P. I., and L. A. Olzak. "Summation of Contrast in Perceived Form from Texture." Perception 26, no. 1_suppl (August 1997): 224. http://dx.doi.org/10.1068/v970121.

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The relative contrast of two superimposed triangles formed by truncated sinusoidal gratings determines which appears more salient. We have previously reported that the saliency of one triangle is selectively enhanced by superimposing a third grating of similar frequency but different orientation. We now ask how the contrasts of the three gratings combine to determine saliency. Stimuli were two superimposed isosceles triangles, formed by overlaying sharply truncated patches of a sinusoidal grating, one at 1.5 cycles deg−1 tilted +45°, the other at 6 cycles deg−1, tilted −45° from vertical. The sharpest-angle apexes pointed in opposite directions (left or right). Contrasts of the gratings were initially adjusted to yield equal performance when observers chose whether the more salient target pointed left or right following a brief (400 ms) monocular exposure. In each test condition a third grating of vertical orientation (spatial frequency 1.5, 3, or 6 cycles deg−1) was added to the entire stimulus at one of six contrast levels ranging from near threshold to 10 × threshold. The point of equal saliency was re-determined from psychometric functions by varying the contrast of one triangle in a 2AFC staircase procedure. The saliency of each triangle was enhanced when the third grating was matched in frequency, but no effect occurred when the third grating differed by an octave (eg was 3 cycles deg−1). Beyond some threshold value of the third grating contrast, the contrast needed to reinstate equal salience was found to be inversely proportional to the contrast of the third grating. Our results are in agreement with the characteristics of higher-level mechanisms that mediate spatial-grain and/or pattern contrast discrimination, and suggest that form-from-texture mechanisms sum component contrasts linearly over a wide range of orientations within a narrow frequency band.

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Bernklau, David. "Verifying a Well-Known Formula." Mathematics Teacher 110, no. 8 (April 2017): 640. http://dx.doi.org/10.5951/mathteacher.110.8.0640.

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One of my favorite lessons, perhaps considered a modernized version of Archimedes's circlecircumference proof, is suitable for students familiar with trigonometric ratios, who know about similar triangles, and who are comfortable using a calculator.

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Journal articles on the topic 'Similar Triangles'

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