Relevant bibliographies by topics / Pythagorean triangles

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Pythagorean triangles'

Author: Grafiati
Published: 3 June 2025
Last updated: 13 July 2025

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Journal articles on the topic "Pythagorean triangles"

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Mita Darbari and Prashans Darbari. "Pythagorean triangles with sum of its two legs as Dodecic." GSC Advanced Engineering and Technology 3, no. 1 (2022): 011–15. http://dx.doi.org/10.30574/gscaet.2022.3.1.0028.

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Number Theory is almost four thousand years old. In ancient clay tablet in Babylon, integral solutions of Pythagorean equations were listed down. Pythagoras theorem has always fascinated young and old mind alike. More than four hundred proofs of the Pythagoras theorem have been discovered so far. Pythagoras theorem is applied in almost every sphere of science- from geometry to Einstein’s theory of relativity. New Pythagorean triangles are being discovered which satisfy certain constraints. Integral solutions of Diophantine equations related to Pythagorean equation are sought by many mathematicians. Applications of these triangles are also being explored in various fields of knowledge. In this paper, we have found nineteen extraordinary Pythagorean Triangles where their sum of two legs are dodecic numbers. These triangles are found by solving the dodecic Diophantine equation using the software Mathematica. Some interesting properties of these Pythagorean Triangles are also observed. Their applications can be explored in cryptography.
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A., Dinesh Kumar, and Vasuki M. "A STUDY ON PYTHAGOREAN TRIPLES." International Journal of Interdisciplinary Research in Arts and Humanities 1, no. 1 (2016): 14–21. https://doi.org/10.5281/zenodo.155337.

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The Pythagorean numbers play a significant role in the theory of higher arithmetic as they come in the majority of indeterminate problem. For the discovery of the law of the three squares (Pythagorean equation), really, one should be indebted to the Pythagorean who were the first Greeks with great intellectual perception. One may notice to his surprise that the Egyptians, the Chinese, the Babylonians and the Indians knew some knowledge of the property of right angled Pythagorean triangles or Pythagorean numbers .Since there is a 1−1 correspondence between Pythagorean numbers and Pythagorean triangles, we shall use them interchangeably. The only geometrical theorem with which the ancient Chinese were acquainted is that the area of the square described on the hypotenuse of a right angled triangle is equal to the sum of the areas of the squares described on the sides. A Pythagorean triangle is a right triangle whose sides are integral lengths.
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Shanthi, J., та M. A. Gopalan. "Formulation of Special Pythagorean Triangles through Integer Solutions of the Hyperbola 𝑦𝟐 = (𝑘𝟐 + 𝟐𝑘)𝑥𝟐 + 𝟏". Indian Journal Of Science And Technology 17, № 41 (2024): 4307–12. http://dx.doi.org/10.17485/ijst/v17i41.3132.

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Objectives: The objective of this research paper is to formulate Pythagorean triangles, each of which satisfies the relation: Hypotenuse = (𝑘 + 1) times the leg with even values added with unity, through employing the integer solutions to the Hyperbola 𝑦2 = (𝑘2 + 2𝑘)𝑥2 + 1. Methods: The sides of the Pythagorean triangle satisfying the requirement are obtained by suitably choosing its generators consisting of the integer solutions to the considered hyperbola. Findings : There are plenty of Pythagorean triangles satisfying the given characterization for each value of k > 0 in the binary quadratic equation given by 𝑦2 = (𝑘2 + 2𝑘)𝑥2 + 1.The characterizations of special pattern of the Pythagorean triangle for 𝑘 = 2is illustrated. Novelty: From the integer solutions of a binary quadratic equation, namely, a hyperbola (Two-dimensional geometrical representation), it is possible to obtain integer solutions to a ternary quadratic equation, namely, Pythagorean triangle (Threedimensional geometrical representation). Keywords: Binary quadratic equation; Ternary quadratic equation; Hyperbola; Pythagorean triangle; Integer solutions
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J, Shanthi, та A. Gopalan M. "Formulation of Special Pythagorean Triangles through Integer Solutions of the Hyperbola 𝑦𝟐 = (𝑘𝟐 + 𝟐𝑘)𝑥𝟐 + 𝟏". Indian Journal of Science and Technology 17, № 41 (2024): 4307–12. https://doi.org/10.17485/IJST/v17i41.3132.

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Abstract <strong>Objectives:</strong>&nbsp;The objective of this research paper is to formulate Pythagorean triangles, each of which satisfies the relation: Hypotenuse = (𝑘 + 1) times the leg with even values added with unity, through employing the integer solutions to the Hyperbola 𝑦2 = (𝑘2 + 2𝑘)𝑥2 + 1.&nbsp;<strong>Methods:</strong>&nbsp;The sides of the Pythagorean triangle satisfying the requirement are obtained by suitably choosing its generators consisting of the integer solutions to the considered hyperbola.&nbsp;<strong>Findings :</strong>&nbsp;There are plenty of Pythagorean triangles satisfying the given characterization for each value of k &gt; 0 in the binary quadratic equation given by 𝑦2 = (𝑘2 + 2𝑘)𝑥2 + 1.The characterizations of special pattern of the Pythagorean triangle for 𝑘 = 2is illustrated.&nbsp;<strong>Novelty:</strong>&nbsp;From the integer solutions of a binary quadratic equation, namely, a hyperbola (Two-dimensional geometrical representation), it is possible to obtain integer solutions to a ternary quadratic equation, namely, Pythagorean triangle (Threedimensional geometrical representation). <strong>Keywords:</strong> Binary quadratic equation; Ternary quadratic equation; Hyperbola; Pythagorean triangle; Integer solutions
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Rajput, Chetansing. "Metallic Ratios in Primitive Pythagorean Triples : Metallic Means embedded in Pythagorean Triangles and other Right Triangles." JOURNAL OF ADVANCES IN MATHEMATICS 20 (July 15, 2021): 312–44. http://dx.doi.org/10.24297/jam.v20i.9088.

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The Primitive Pythagorean Triples are found to be the purest expressions of various Metallic Ratios. Each Metallic Mean is epitomized by one particular Pythagorean Triangle. Also, the Right Angled Triangles are found to be more “Metallic” than the Pentagons, Octagons or any other (n2+4)gons. The Primitive Pythagorean Triples, not the regular polygons, are the prototypical forms of all Metallic Means.
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Hikmatovich, Ibragimov Husniddin. "Connection Between A Right Triangle And An Equal Side Triangle." American Journal of Interdisciplinary Innovations and Research 02, no. 11 (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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Swenson, Carl, and André Yandl. "Partitioning Pythagorean Triangles Using Pythagorean Angles." College Mathematics Journal 43, no. 3 (2012): 220–25. http://dx.doi.org/10.4169/college.math.j.43.3.220.

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Choi, Eunmi. "FIBONACCI TYPE PYTHAGOREAN TRIANGLES." Far East Journal of Mathematical Sciences (FJMS) 101, no. 9 (2017): 2023–41. http://dx.doi.org/10.17654/ms101092023.

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S., Yahya Mohamed *. &. M. Prem Kumar. "A NOTE ON SPECIAL PAIRS OF PYTHAGOREAN TRIANGLE AND 3-DIGIT SPHENIC NUMBER." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 6, no. 7 (2017): 242–46. https://doi.org/10.5281/zenodo.823122.

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In this paper , we present pairs of Pythagorean triangles such in each pair, the difference between their perimeters is four times the 3-digit Sphenic number 110. Also we present the number of pairs of primitive and non- primitive Pythagorean triangles.
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S., Yahya Mohamed *1 S. Prem Kumar 2. and V. Somu3. "SPECIAL PAIRS OF PYTHAGOREAN TRIANGLES AND 3 –DIGITS HARSHAD NUMBER." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 6, no. 8 (2017): 226–29. https://doi.org/10.5281/zenodo.843869.

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Here, we present pairs of Pythagorean triangles such that in each pair, the difference between their perimeters is four times the 3-digit Harshad number 108. Also we present the number of pairs of primitive and non- primitive Pythagorean triangles.
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Dissertations / Theses on the topic "Pythagorean triangles"

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Wigren, Thomas. "The Cauchy-Schwarz inequality : Proofs and applications in various spaces." Thesis, Karlstads universitet, Avdelningen för matematik, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-38196.

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We give some background information about the Cauchy-Schwarz inequality including its history. We then continue by providing a number of proofs for the inequality in its classical form using various proof techniques, including proofs without words. Next we build up the theory of inner product spaces from metric and normed spaces and show applications of the Cauchy-Schwarz inequality in each content, including the triangle inequality, Minkowski's inequality and Hölder's inequality. In the final part we present a few problems with solutions, some proved by the author and some by others.
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Jesus, Manoel Bernardes. "Triângulo: formas, medidas e aplicações." Universidade Federal de Goiás, 2016. http://repositorio.bc.ufg.br/tede/handle/tede/5639.

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Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2016-06-02T18:51:04Z No. of bitstreams: 2 Dissertação - Manoel Bernardes de Jesus- 2016.pdf: 8498646 bytes, checksum: df9ec5bab5a7bc205f3b14b7ac2d5b70 (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5)<br>Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2016-06-03T12:08:15Z (GMT) No. of bitstreams: 2 Dissertação - Manoel Bernardes de Jesus- 2016.pdf: 8498646 bytes, checksum: df9ec5bab5a7bc205f3b14b7ac2d5b70 (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5)<br>Made available in DSpace on 2016-06-03T12:08:15Z (GMT). No. of bitstreams: 2 Dissertação - Manoel Bernardes de Jesus- 2016.pdf: 8498646 bytes, checksum: df9ec5bab5a7bc205f3b14b7ac2d5b70 (MD5) license_rdf: 19874 bytes, checksum: 38cb62ef53e6f513db2fb7e337df6485 (MD5) Previous issue date: 2016-03-31<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES<br>The aim of this paper is to present a study about Triangles, to study their concepts and their properties, the relations between their forms, mesures and areas, relation of Triangle with the Trigonometric and the Geometrics Builds. Besides to present Pythagoras’ Theorem with many proof relates to Triangles.<br>O objetivo deste trabalho é apresentar um estudo sobre Triângulos, seus conceitos e suas propriedades, as relações entre suas formas, medidas e áreas. Além da relação do Triângulo com a Trigonometria e as Construções Geométricas. Apresentamos o Teorema de Pitágoras com várias demonstrações relacionadas aos Triângulos.
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Leite, Adriane de Oliveira. "Material complementar para o professor da rede SESI-SP de ensino : semelhança e software GeoGebra." Universidade Federal de São Carlos, 2015. https://repositorio.ufscar.br/handle/ufscar/7578.

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Submitted by Daniele Amaral (daniee_ni@hotmail.com) on 2016-09-21T19:25:25Z No. of bitstreams: 1 DissAOL.pdf: 6820225 bytes, checksum: 00793d2a933ccdc18ce28c12bf53f7ac (MD5)<br>Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-09-28T19:38:02Z (GMT) No. of bitstreams: 1 DissAOL.pdf: 6820225 bytes, checksum: 00793d2a933ccdc18ce28c12bf53f7ac (MD5)<br>Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-09-28T19:38:13Z (GMT) No. of bitstreams: 1 DissAOL.pdf: 6820225 bytes, checksum: 00793d2a933ccdc18ce28c12bf53f7ac (MD5)<br>Made available in DSpace on 2016-09-28T19:43:11Z (GMT). No. of bitstreams: 1 DissAOL.pdf: 6820225 bytes, checksum: 00793d2a933ccdc18ce28c12bf53f7ac (MD5) Previous issue date: 2015-10-05<br>Não recebi financiamento<br>This research aims to propose activities for teachers using the Geogebra software, especially for teachers from the SESI-SP School Network in order to assist them in the teaching methodology, with teachers' work plan and, in addition, aiming to more significant and dynamic classes, in order to allow students reach their teaching and learning expectations, formulate valid arguments, make conjectures and justify their reasoning. The activities were applied by teachers of SESI-SP School Network to the students of 9th grade of elementary school, in anticipation of teaching and learning through “Similarity”, addressing Theorem of Thales, Metrics Relations in the Rectangle Triangle and Pythagoras Theorem. The results were analyzed and discussed, reporting the challenges and conclusions raised by the students during the activities while working with the Geogebra software and also based on the feedback provided by the teachers and the opinion of the analysts from SESI-SP School Network.<br>Esta pesquisa tem como objetivo principal propor atividades para os professores utilizando o software Geogebra, principalmente para os docentes da rede SESI-SP de Ensino, a fim de auxiliá-los na metodologia de ensino, no plano de trabalho, visando uma aula mais significativa e dinâmica, para que seus alunos atinjam as expectativas de ensino e aprendizagem, formulem argumentos válidos, façam conjecturas e justifiquem seus raciocínios. As atividades foram aplicadas por professores da rede SESI-SP de Ensino aos alunos do 9º ano do Ensino Fundamental, turma de 2014, na expectativa de ensino e aprendizagem de “Semelhança”, abordando Teorema de Tales, Relações Métricas no Triângulo Retângulo e Teorema de Pitágoras. Os resultados foram analisados e discutidos, relatando as dificuldades e conclusões apresentadas pelos alunos em desenvolver as atividades trabalhando com o software Geogebra, baseado nas devolutivas dos professores envolvidos e o parecer feito pelos analistas educacionais da Rede SESI-SP de Ensino.
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Lau, Christina 1987. "Pythagorean theorem extensions." Thesis, 2011. http://hdl.handle.net/2152/ETD-UT-2011-08-3839.

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This report expresses some of the recent research surrounding the Pythagorean Theorem and Pythagorean triples. Topics discussed include applications of the Pythagorean Theorem relating to recursion methods, acute and obtuse triangles, Pythagorean triangles in squares, as well as Pythagorean boxes. A short discussion on the depth of the Pythagorean Theorem taught in secondary schools is also included.<br>text
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Books on the topic "Pythagorean triangles"

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Sierpiński, Wacław. Pythagorean triangles. Dover Publications, 2003.

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Sierpinski, Waclaw. Pythagorean Triangles. Dover Publications, Incorporated, 2013.

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Sierpinski, Waclaw. Pythagorean Triangles. Dover Publications, Incorporated, 2013.

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Sierpiński, Wacław. Pythagorean Triangles. Dover Publications, 2003.

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The Pythagorean Theorem & Right Triangles. 2015.

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Takloo-Bighash, Ramin. Pythagorean Introduction to Number Theory: Right Triangles, Sums of Squares, and Arithmetic. Springer International Publishing AG, 2021.

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Takloo-Bighash, Ramin. A Pythagorean Introduction to Number Theory: Right Triangles, Sums of Squares, and Arithmetic. Springer, 2018.

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Metaphysics of the Pythagorean Theorem: Thales, Pythagoras, Engineering, Diagrams, and the Construction of the Cosmos Out of Right Triangles. State University of New York Press, 2017.

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Joujan, Alex. Summit Math Algebra 2 Book 8: The Pythagorean Theorem and Special Right Triangles. Independently Published, 2020.

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Metaphysics of the Pythagorean Theorem: Thales, Pythagoras, Engineering, Diagrams, and the Construction of the Cosmos Out of Right Triangles. State University of New York Press, 2018.

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Book chapters on the topic "Pythagorean triangles"

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Anglin, W. S., and J. Lambek. "Pythagorean Triangles and Fermat’s Last Theorem." In The Heritage of Thales. Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4612-0803-7_50.

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Takloo-Bighash, Ramin. "What integers are areas of right triangles?" In A Pythagorean Introduction to Number Theory. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-02604-2_4.

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Enfield, Jacob. "Distance | Right Triangles, Pythagorean Theorem, and the Distance Formula." In Mathematics of Game Development. CRC Press, 2024. http://dx.doi.org/10.1201/9781032701431-5.

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Takloo-Bighash, Ramin. "What numbers are the edges of a right triangle?" In A Pythagorean Introduction to Number Theory. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-02604-2_5.

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"Pythagorean Triangles." In A Guide to Elementary Number Theory. American Mathematical Society, 2009. http://dx.doi.org/10.5948/upo9780883859186.021.

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Pickover, Clifford A. "Everything You Wanted to Know about Triangles but Were Afraid to Ask." In Wonders of Numbers. Oxford University PressNew York, NY, 2001. http://dx.doi.org/10.1093/oso/9780195133424.003.0099.

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Abstract Pythagorean triangles with integral sides have been the subject of a huge amount of mathematical inquiry. For example, Albert Beiler, author of Recreations in the Theory of Numbers, has been interested in Pythagorean triangles with large consecutive leg values. These triangles are as rare as diamonds for small legs. Triangle 3-4-5 is the first of these exotic gems. The next such one is 21-20-29. The tenth such triangle is quite large: 27304197-27304196-38613965.
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Nicolaides, Demetris. "Numbers and Shapes." In In Search of a Theory of Everything. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780190098353.003.0006.

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Pythagoras initiated the mathematical analysis of nature, a cornerstone practice in modern physics. “Things are numbers” is the most significant Pythagorean doctrine. It signifies that the phenomena of nature are describable by equations and numbers. Therefore, nature is quantifiable and potentially knowable through the scientific method. The Pythagoreans quantified pleasing sounds of music, right-angled triangles, even the motion of the heavenly bodies. The “Copernican revolution” (heliocentricity) is traced back to Pythagorean cosmology. But, finally, Einstein’s relativity clarifies a popular misconception related to it: that “the earth revolves around the sun (heliocentricity) is correct,” and that “the sun revolves around the earth (geocentricism) is incorrect.” Plato was inspired by Pythagorean mathematics, but he replaced “things are numbers” with things are shapes, forms, Forms, a noetic description of nature known as the theory of “Forms.” The quantum-mechanical wave-functions—mathematical forms that describe microscopic particles—are the Platonic Forms of quarks and leptons.
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Van Brummelen, Glen. "The Modern Approach: Right-Angled Triangles." In Heavenly Mathematics. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691175997.003.0005.

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This chapter discusses the modern approach to solving right-angled triangles. After a brief background on John Napier's trigonometric work, in which he referred mostly to right-angled spherical triangles, the chapter describes the theorems for right triangles. It then considers an oblique triangle split into two right triangles and the ten fundamental identities of a right-angled spherical triangle, how the locality principle can be applied to derive the Pythagorean Theorem, and how to find a ship's direction of travel using the theorem. It also looks at Napier's work on logarithms which was devoted to trigonometry, along with Napier's Rules. The chapter concludes with an overview of “pentagramma mirificum,” a pentagram in spherical trigonometry that was discovered by Napier.
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Van Brummelen, Glen. "The Modern Approach: Oblique Triangles." In Heavenly Mathematics. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691175997.003.0006.

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This chapter discusses the modern approach to solving oblique triangles. Two important theorems about planar oblique triangles are the spherical and planar Law of Sines and the Law of Cosines, which is an extension of the Pythagorean Theorem applied to oblique triangles. Book I of Euclid's Elements deals primarily with the Pythagorean Theorem (Proposition 47) and its converse (Proposition 48), while Book II contains theorems that may be translated directly into various algebraic statements. The chapter considers two of the last three theorems of Book II: Proposition 12, which deals with obtuse-angled triangles, and Proposition 13, which is concerned with acute-angled triangles. It also extends the Law of Cosines to the sphere and uses it to solve astronomical and geographical problems, such as finding the distance from Vancouver to Edmonton. Finally, it describes Delambre's analogies and Napier's analogies.
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Ayyaswamy, Kathirvel, A. Anthony Raj, S. Prathap, D. Madhavakumar, A. Nelson, and Naren Kathirvel. "An Application of Pythagoras Theorem From Heron's Formula to Derive the Foci of an Ellipse." In Advances in Computational Intelligence and Robotics. IGI Global, 2025. https://doi.org/10.4018/979-8-3693-8985-0.ch004.

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In this paper, we derived the Pythagoras theorem from Heron's formula. Also we provide the application of this result plays vital role to obtain foci of an ellipse. The foci of an ellipse are derived through a combination of geometric principles. While Heron's formula is useful for calculating areas of triangles, the primary derivation of the foci comes from the Pythagorean theorem, which connects the semi-major and semi-minor axes of the ellipse. The relationship between the Pythagorean theorem, Heron's formula, and the foci of an ellipse is an interesting geometric application. By offering personalized, adaptable learning, instant feedback, and interactive experiences, AI and humanoid computing empower cyber nomads to stay engaged, progress at their own pace, and master subjects effectively. These technologies remove barriers of time and location, creating a learning environment that is as responsive and supportive as any traditional classroom. For cyber nomads, AI and humanoid computing represent a lifeline to continuous, high-quality education no matter where they are in the world.
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Conference papers on the topic "Pythagorean triangles"

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Darbari, Mita, and Prashans Darbari. "Eight extraordinary pythagorean triangles." In INTELLIGENT BIOTECHNOLOGIES OF NATURAL AND SYNTHETIC BIOLOGICALLY ACTIVE SUBSTANCES: XIV Narochanskie Readings. AIP Publishing, 2023. http://dx.doi.org/10.1063/5.0179381.

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Darbari, Mita, and Prashans Darbari. "Connection between primitive Pythagorean triangles and Mersenne primes." In FOURTH INTERNATIONAL CONFERENCE ON ADVANCES IN PHYSICAL SCIENCES AND MATERIALS: ICAPSM 2023. AIP Publishing, 2024. http://dx.doi.org/10.1063/5.0216176.

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Magnaghi-Delfino, Paola, Giampiero Mele, and Tullia Norando. "Il pentagono come strumento per il disegno delle fortezze." In FORTMED2020 - Defensive Architecture of the Mediterranean. Universitat Politàcnica de València, 2020. http://dx.doi.org/10.4995/fortmed2020.2020.11324.

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The pentagon as a tool for fortresses’ drawingStarting from the fifteenth century, the diagram of many fortresses has a pentagonal shape. Among the best known fortresses, in Italy we find the Fortezza da Basso of Florence, the Cittadella of Parma, the Cittadella of Turin, Castel Sant’Angelo in Rome. The aim of this article is to analyze the reasons that link form and geometry to the planning of the design and the layout of pentagonal fortresses. The pentagon is a polygon tied to the golden section and to the Fibonacci sequence and it is possible to construct it starting from the golden triangle and its gnomon. This construction of the pentagon is already found in the book De Divina Proportione by Luca Pacioli and is particularly convenient for planning pentagonal fortresses. If one wants to draw the first approximated golden triangle, one can just consider the numbers of the Fibonacci sequence, for example 5 and 8, which establish the relationship between the sides: 5 units is the length of the base and 8 units the length of the equal sides. In the second isosceles triangle, which is the gnomon of the first, the base is 8 units long and equal sides are 5 units long; half of this isosceles triangle is the Pythagorean triangle (3, 4, 5). This characteristic of the golden triangles, that was already known by the Pythagoreans and, in a certain sense, contained in the symbol of their School, allows to build a pentagon with only the use of the ruler and the set square. The distinctive trait of the construction just described makes preferable to use the pentagon in the layout of the military architectures in the fieldworks. We have verified the relationship between numbers, shape and size in the layout of Castel Sant’Angelo (1555-1559) in which the approximate pentagon was the instrument for the generation of its form.
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Latinčić, Dragan. "Experiments with Rhythm Using the Method of Harmonic Spectrum Projection." In Iskošeni ugao Dragutina Gostuškog. Muzikološki institut SANU Beograd, 2024. https://doi.org/10.46793/dgost23.129l.

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In his capital work The Time of Art. The Contribution to the Foundation of a General Science of Forms, Dragutin Gostuški also points to the perspective of musical art, seen through significant changes in the theory of rhythm and the theory of proportions. Gostuški is a follower of the Pythagorean teachings on music, which he accepted creatively, as he looked for beauty and harmony in formulas, mathematical expressions, and laws of physics. This is mostly manifested in his discussions and postulates on rhythm. Using numerous and valuable examples, Gostuški points to an extraordinary synthetic power in the field of rhythm theory and rhythmic phenomena by employing either a direct relevant method or a comparative method through the prism of the exact scientific field that describes the given phenomenon. In Gostuški’s book, numbers and shapes (rhythmic and geometric), as mathematical objects, found a natural refuge in scientific rigor, whereas they embodied spiritual existence as an autonomous creative act in communication with the reader. This is mediated by Gostuški’s method of projecting a series of harmonics, which the author introduces by explaining the natural connection of vertical and horizontal sound entities. Reading Gostuški’s texts, the reader spontaneously comes to the conclusion that the accents of rhythmic groups, by means of geometry, can be identified with reference points, and that their mutual relations can be represented geometrically by polygonal lines, which allows us to meet triangle, square, pentagon, cube, and hypercube, again in a new way, which would give new confirmation to Pythagoras’ and Plato’s visions. In the text, I included a short section from my composition Inflections for piano, dedicated to our fellow composer and pianist Stanko Simić, who commissioned the composition, and through which I will spontaneously cover both the analytical and the compositional-practical aspects of the phenomenon of rhythmic projection. Gostuški’s dialectic is challenging because it is revealing and unique, and some of his postulates can also inspire us to derive a system of compositional algorithms that are naturally incorporated into creative poetics.
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Kumar, Vineet. "On the dynamic behaviour of pythagorean triangle employed in power space on account of bilateral swing for given hypotenuse arm." In 2018 2nd International Conference on Inventive Systems and Control (ICISC). IEEE, 2018. http://dx.doi.org/10.1109/icisc.2018.8399005.

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