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Journal articles on the topic 'Ceva's theorem'

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Published: 4 June 2021

Last updated: 25 April 2022

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1

Contreras, José N. "Discovering, Applying, and Extending Ceva's Theorem." Mathematics Teacher 108, no. 8 (April 2015): 632–37. http://dx.doi.org/10.5951/mathteacher.108.8.0632.

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2

Hoehn, Larry. "73.21 A Simple Generalisation of Ceva's Theorem." Mathematical Gazette 73, no. 464 (June 1989): 126. http://dx.doi.org/10.2307/3619672.

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3

Landy, Steven. "A Generalization of Ceva's Theorem to Higher Dimensions." American Mathematical Monthly 95, no. 10 (December 1988): 936. http://dx.doi.org/10.2307/2322390.

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4

Landy, Steven. "A Generalization of Ceva's Theorem to Higher Dimensions." American Mathematical Monthly 95, no. 10 (December 1988): 936–39. http://dx.doi.org/10.1080/00029890.1988.11972122.

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5

Koichu, Boris, and Abraham Berman. "3-D Dynamic Geometry: Ceva's Theorem in Space." International Journal of Computers for Mathematical Learning 9, no. 1 (2004): 95–108. http://dx.doi.org/10.1023/b:ijco.0000038277.75552.65.

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6

Soydan, Gökhan, Yusuf Doğru, and Umut Arslandoğan. "On the ratio of directed lengths on the plane with generalized absolute value metric and related properties." Filomat 26, no. 1 (2012): 119–29. http://dx.doi.org/10.2298/fil1201119s.

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In this paper, we show that a point of division divides a related line segment in the same ratio on the plane with generalized absolute value metric and Euclidean plane. Then the coordinates of the division point can be determined by the same formula as in the Euclidean plane. In the latter parts of the work, we give Ceva's and Menelaus'es theorems and the theorem of directed lines on the plane with generalized absolute value metric.

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7

Byrkit, Donald R., and Timothy L. Dixon. "A Corollary to Ceva's Theorem and Some of Its Consequences." School Science and Mathematics 90, no. 8 (December 1990): 683–93. http://dx.doi.org/10.1111/j.1949-8594.1990.tb12047.x.

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8

Annersih, Nevi, Mashadi Mashadi, and M. D. H. Gamal. "PENGEMBANGAN TEOREMA CEVA PADA SEGILIMA." JURNAL MATHEMATIC PAEDAGOGIC 3, no. 1 (July 10, 2018): 47. http://dx.doi.org/10.36294/jmp.v3i1.309.

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Abstract This paper discusses the development of the Ceva’s theorem on the pentagon in various cases including for the convex pentagon and the nonconvent pentagon. The Ceva’s theorem discusses the case of one-point concurrent in the pentagon. The proofing process is done in a simple way that is by using wide comparison. The results obtained from this paper are the existence of five lines from each vertex on the pentagon intersected at one point (concurrent) ie point P. Keywords: Ceva theorem, Ceva’s theorem on the pentagon, concurrent AbstrakTulisan ini membahas tentang pengembangan teorema Ceva pada segilima dalam berbagai kasus antara lain untuk segilima konveks dan segilima nonkonveks. Teorema Ceva segilima membahas kasus kekonkurenan satu titik yang berada pada segilima. Proses pembuktian dilakukan dengan cara sederhana yaitu dengan menggunakan perbandingan luas. Hasil yang diperoleh dari tulisan ini adalah eksistensi lima buah garis dari masing-masing titik sudut pada segilima berpotongan di satu titik (konkuren) yaitu titik P. Kata kunci: teorema Ceva, teorema Ceva pada segilima, konkurensi

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9

Coghetto, Roland. "Altitude, Orthocenter of a Triangle and Triangulation." Formalized Mathematics 24, no. 1 (March 1, 2016): 27–36. http://dx.doi.org/10.1515/forma-2016-0003.

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Summary We introduce the altitudes of a triangle (the cevians perpendicular to the opposite sides). Using the generalized Ceva’s Theorem, we prove the existence and uniqueness of the orthocenter of a triangle [7]. Finally, we formalize in Mizar [1] some formulas [2] to calculate distance using triangulation.

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10

Jupri, Al, Siti Fatimah, and Dian Usdiyana. "Dampak Perkuliahan Geometri Pada Penalaran Deduktif Mahasiswa: Kasus Pembelajaran Teorema Ceva." AKSIOMA : Jurnal Matematika dan Pendidikan Matematika 11, no. 1 (July 15, 2020): 93–104. http://dx.doi.org/10.26877/aks.v11i1.6011.

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Geometry is one of branches of mathematics that can develop deductive thinking ability for anyone, including students of prospective mathematics teachers, who learning it. This deductive thinking ability is needed by prospective mathematics teachers for their future careers as mathematics educators. This research therefore aims to investigate the influence of the learning process of a geometry course toward deductive reasoning ability of students of prospective mathematics teachers. To do so, this qualitative research was carried out through an observation of the learning process and assessment of the geometry course, involving 56 students of prospective mathematics teachers, in one of mathematics education program, in one of state universities in Bandung. A geometry topic observed in the learning process was the Ceva’s theorem, and the assessment was in the form of an individual written test on the application of the Ceva’s theorem in a proving process. The results showed that the learning process emphasizes on proving of theorems and mathematical statements. In addition, the test revealed that ten students are able to use the Ceva’s theorem in a proving process and different strategies of proving are found, including the use of properties of similarity between triangles and of the concept of trigonometry. This indicates a creativity of student deductive thinking in proving process. In conclusion, the geometry course that emphasizes on proving of theorems and mathematical statements has influenced on filexibility of student deductive thinking in proving processes.

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11

Shminke, Boris A. "Routh’s, Menelaus’ and Generalized Ceva’s Theorems." Formalized Mathematics 20, no. 2 (December 1, 2012): 157–59. http://dx.doi.org/10.2478/v10037-012-0018-9.

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Summary The goal of this article is to formalize Ceva’s theorem that is in the [8] on the web. Alongside with it formalizations of Routh’s, Menelaus’ and generalized form of Ceva’s theorem itself are provided.

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12

Coghetto, Roland. "Circumcenter, Circumcircle and Centroid of a Triangle." Formalized Mathematics 24, no. 1 (March 1, 2016): 17–26. http://dx.doi.org/10.1515/forma-2016-0002.

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Summary We introduce, using the Mizar system [1], some basic concepts of Euclidean geometry: the half length and the midpoint of a segment, the perpendicular bisector of a segment, the medians (the cevians that join the vertices of a triangle to the midpoints of the opposite sides) of a triangle. We prove the existence and uniqueness of the circumcenter of a triangle (the intersection of the three perpendicular bisectors of the sides of the triangle). The extended law of sines and the formula of the radius of the Morley’s trisector triangle are formalized [3]. Using the generalized Ceva’s Theorem, we prove the existence and uniqueness of the centroid (the common point of the medians [4]) of a triangle.

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13

Klamkin, Murray S., and Sidney H. Kung. "Ceva's and Menelaus' Theorems and Their Converses via Centroids." Mathematics Magazine 69, no. 1 (February 1, 1996): 49. http://dx.doi.org/10.2307/2691397.

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14

Klamkin, Murray S., and Sidney H. Kung. "Ceva's and Menelaus' Theorems and Their Converses via Centroids." Mathematics Magazine 69, no. 1 (February 1996): 49–51. http://dx.doi.org/10.1080/0025570x.1996.11996382.

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15

Carden, Russell, and Derek J. Hansen. "Ritz values of normal matrices and Ceva’s theorem." Linear Algebra and its Applications 438, no. 11 (June 2013): 4114–29. http://dx.doi.org/10.1016/j.laa.2012.12.030.

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16

Samet, Dov. "An Extension of Ceva’s Theorem to n-Simplices." American Mathematical Monthly 128, no. 5 (April 27, 2021): 435–45. http://dx.doi.org/10.1080/00029890.2021.1896292.

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17

Grunbaum, Branko, and G. C. Shephard. "A New Ceva-Type Theorem." Mathematical Gazette 80, no. 489 (November 1996): 492. http://dx.doi.org/10.2307/3618512.

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18

Kozma, József, and Árpád Kurusa. "Ceva’s and Menelaus’ theorems characterize the hyperbolic geometry among Hilbert geometries." Journal of Geometry 106, no. 3 (December 7, 2014): 465–70. http://dx.doi.org/10.1007/s00022-014-0258-7.

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19

Bradley, Christopher J. "Hexagons with opposite sides parallel." Mathematical Gazette 90, no. 517 (March 2006): 57–67. http://dx.doi.org/10.1017/s0025557200179033.

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This paper presents a number of theorems about hexagons whose three pairs of opposite sides are parallel. The first of these is a well-known result that the vertices of such a hexagon lie on a conic. Theorems 3 and 4 show how such conies are related to the Cevians of a triangle, and which Cevians lead to such conies being circles. When they are circles they are called Tucker circles. None of the results is at all obvious, yet it seems that some of the results presented here were known in the late nineteenth century or the early twentieth century. They seem to be of more than passing interest which should not get lost by the passage of time.

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20

Hoehn, Larry. "89.49 A Ceva-type theorem for the cyclic quadrilateral." Mathematical Gazette 89, no. 515 (July 2005): 282–83. http://dx.doi.org/10.1017/s0025557200177848.

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21

Klamkin, Murray S., and Andy Liu. "Simultaneous Generalizations of the Theorems of Ceva and Menelaus." Mathematics Magazine 65, no. 1 (February 1, 1992): 48. http://dx.doi.org/10.2307/2691362.

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22

Klamkin, Murray S., and Andy Liu. "Simultaneous Generalizations of the Theorems of Ceva and Menelaus." Mathematics Magazine 65, no. 1 (February 1992): 48–52. http://dx.doi.org/10.1080/0025570x.1992.11995977.

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23

Man, Y. K. "A simple proof of the generalized Ceva theorem by the principle of equilibrium." International Journal of Mathematical Education in Science and Technology 38, no. 4 (June 15, 2007): 566–69. http://dx.doi.org/10.1080/00207390701240794.

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24

Su, Stephen, and Cheng Shyong Lee. "Simultaneous Generalizations of the Theorems of Menelaus, Ceva, Routh, and Klamkin/Liu." Mathematics Magazine 91, no. 4 (August 8, 2018): 294–303. http://dx.doi.org/10.1080/0025570x.2018.1495435.

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25

Houston, Kelly B., and Robert C. Powers. "Simultaneous generalizations of the theorems of Ceva and Menelaus for field planes." International Journal of Mathematical Education in Science and Technology 40, no. 8 (December 15, 2009): 1085–91. http://dx.doi.org/10.1080/00207390903121776.

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26

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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27

Laflamme, Jeanne, and Matilde Lalín. "On Ceva points of (almost) equilateral triangles." Journal of Number Theory 222 (May 2021): 48–74. http://dx.doi.org/10.1016/j.jnt.2020.12.007.

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28

Merdrignac, Bernard. "Marie-Madeleine de Cevins et Jean-Michel Matz, Formation intellectuelle et culture du clergé dans les territoires angevins (milieu du XIIIe-fin du XVe siècle)." Annales de Bretagne et des pays de l'Ouest, no. 113-2 (June 30, 2006): 191–93. http://dx.doi.org/10.4000/abpo.850.

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29

Jakobsen, Johnny Grandjean Gøgsig. "Marie-Madeleine de Cevins, Confraternity, Mendicant Orders, and Salvation in the Middle Ages: The Contribution of the Hungarian Sources (c.1270–c.1530). (Europa Sacra 23.) Turnhout: Brepols, 2018. Pp. xvii, 375; 22 black-and-white plates, 2 maps, 14 graphs, and 2 tables. €100. ISBN: 978-2-5035-7871-2." Speculum 95, no. 4 (October 1, 2020): 1146–47. http://dx.doi.org/10.1086/710645.

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30

Smarandache, Florentin. "An Application of the Generalization of Ceva's Theorem." SSRN Electronic Journal, 2016. http://dx.doi.org/10.2139/ssrn.2725515.

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31

Akça, Ziya, and Selahattin Nazlı. "On the versions in the plane $\mathbb{R}_{\pi 3}^{2}$ of some Euclidean theorems." New Trends in Mathematical Science, March 31, 2022. http://dx.doi.org/10.20852/ntmsci.2022.474.

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In this paper, we give the ratios of directed lengths\ in planes the Euclidean and $\mathbb{R}_{\pi 3}^{2}$ and the analogues in the plane $ \mathbb{R}_{\pi 3}^{2}$ of the Menelaus and Ceva's Theorems.

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32

Witczyński, Krzysztof. "CEVA’S AND MENELAUS’ THEOREMS FOR TETRAHEDRA (II)." Demonstratio Mathematica 29, no. 1 (January 1, 1996). http://dx.doi.org/10.1515/dema-1996-0127.

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33

Ushijima, Akira. "Ceva’s and Menelaus’ theorems for higher-dimensional simplexes." Journal of Geometry 110, no. 1 (February 6, 2019). http://dx.doi.org/10.1007/s00022-019-0468-0.

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34

Kurusa, Árpád. "Ceva’s and Menelaus’ theorems in projective-metric spaces." Journal of Geometry 110, no. 2 (July 12, 2019). http://dx.doi.org/10.1007/s00022-019-0495-x.

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