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Journal articles on the topic 'HERONIAN TRIANGLE'

Author: Grafiati

Published: 4 June 2021

Last updated: 4 February 2022

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1

Dolan, Stan. "Ratios in Heronian triangles." Mathematical Gazette 104, no. 560 (June 18, 2020): 193–208. http://dx.doi.org/10.1017/mag.2020.41.

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Heronian triangles are triangles with integer sides and area. In [1], a classical result about squares in arithmetic progression was obtained by proving that it is not possible for the altitude of a Heronian triangle to divide the base in the ratio of 1 : 2. In this article we shall investigate general ratios of the form 1 : n, where n is a positive integer. These base ratios do not appear to have received previous attention despite the wealth of results about other aspects of Heronian triangles [2].

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2

Mazėtis, Edmundas, and Grigorijus Melničenko. "Rational cuboids and Heron triangles II." Lietuvos matematikos rinkinys 60 (December 5, 2019): 34–38. http://dx.doi.org/10.15388/lmr.b.2019.15233.

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We study the connection of Heronian triangles with the problem of the existence of rational cuboids. It is proved that the existence of a rational cuboid is equivalent to the existence of a rectangular tetrahedron, which all sides are rational and the base is a Heronian triangle. Examples of rectangular tetrahedra are given, in which all sides are integer numbers, but the area of the base is irrational. The example of the rectangular tetrahedron is also given, which has lengths of one side irrational and the other integer, but the area of the base is integer.

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3

Mazėtis, Edmundas, and Grigorijus Melničenko. "Algebraic values of sines and cosines and their arguments." Lietuvos matematikos rinkinys 61 (March 15, 2021): 21–28. http://dx.doi.org/10.15388/lmr.2020.22717.

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The article introduces the reader to some amazing properties of trigonometric functions. It turns out that if the values of the arguments of the functions sin x, cos x, tg x and ctg x, expressed in radians, are algebraic numbers, then the values of these functions are transcendental numbers. Hence, it follows that the values of all angles of the pseudo-Heronian triangle, including the values of all angles of the Pythagoras or Heron triangle, expressed in radians, are transcendental numbers. If the arguments of functions sin x and cos x, expressed in radians, are equal to x = r 2 \pi, where r are rational numbers, then the values of the functions are algebraic numbers. It should be noted that in this case the argument x = r 2\pi is transcendental and, if expressed in degrees, becomes a rational.

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4

Read, Emrys. "100.01 Heronian triangles." Mathematical Gazette 100, no. 547 (March 2016): 103–8. http://dx.doi.org/10.1017/mag.2016.10.

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5

Yiu, Paul. "Heronian Triangles Are Lattice Triangles." American Mathematical Monthly 108, no. 3 (March 2001): 261. http://dx.doi.org/10.2307/2695390.

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6

Yiu, Paul. "Heronian Triangles Are Lattice Triangles." American Mathematical Monthly 108, no. 3 (March 2001): 261–63. http://dx.doi.org/10.1080/00029890.2001.11919751.

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7

Stephenson, Paul. "92.55 Reconstructing heronian triangles." Mathematical Gazette 92, no. 524 (July 2008): 328–31. http://dx.doi.org/10.1017/s0025557200183342.

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8

Nelsen, Roger B. "Almost Equilateral Heronian Triangles." Mathematics Magazine 93, no. 5 (October 19, 2020): 378–79. http://dx.doi.org/10.1080/0025570x.2020.1817708.

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9

Kozhegel'dinov, S. Sh. "On fundamental Heronian triangles." Mathematical Notes 55, no. 2 (February 1994): 151–56. http://dx.doi.org/10.1007/bf02113294.

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10

Dolan, Stan. "Less than equable Heronian triangles." Mathematical Gazette 100, no. 549 (October 17, 2016): 482–89. http://dx.doi.org/10.1017/mag.2016.113.

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It is well known that there are precisely five integer-sided triangles which have equal area, Δ, and perimeter, P. These triangles are called equable Heronian triangles.A proof of this result was given by Whitworth [1]. Since Whitworth's time, much attention has been given to triangles whose areas are integer multiples of their perimeters, for example [2, 3]. However, as this paper will show, Heronian triangles with areas less than their perimeters have some mathematical interest.

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11

Yiu, Paul. "Construction of Indecomposable Heronian Triangles." Rocky Mountain Journal of Mathematics 28, no. 3 (September 1998): 1189–202. http://dx.doi.org/10.1216/rmjm/1181071762.

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12

McLean, K. Robin. "72.13 Heronian Triangles Are Almost Everywhere." Mathematical Gazette 72, no. 459 (March 1988): 49. http://dx.doi.org/10.2307/3617996.

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13

Dolan, Stan. "103.29 Heronian triangles and squares in arithmetic progression." Mathematical Gazette 103, no. 558 (October 21, 2019): 490–93. http://dx.doi.org/10.1017/mag.2019.111.

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14

Arimoto, Koichi, and Yasuyuki Hirano. "ON LATTICE POINTS WHICH BECOME VERTICES OF HERONIAN TRIANGLES." Far East Journal of Mathematical Sciences (FJMS) 107, no. 2 (October 11, 2018): 511–18. http://dx.doi.org/10.17654/ms107020511.

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15

Bailey, Herb, and William Gosnell. "Heronian Triangles with Sides in Arithmetic Progression: An Inradius Perspective." Mathematics Magazine 85, no. 4 (October 2012): 290–94. http://dx.doi.org/10.4169/math.mag.85.4.290.

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16

Arimoto, Koichi, and Yasuyuki Hirano. "ON LATTICE POINTS WHICH BECOME VERTICES OF HERONIAN TRIANGLES II." Far East Journal of Mathematical Sciences (FJMS) 129, no. 2 (April 1, 2021): 157–59. http://dx.doi.org/10.17654/ms129020157.

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17

Mazėtis, Edmundas, and Grigorijus Melničenko. "A rational sine and cosine of the angles of a triangle." Lietuvos matematikos rinkinys 54 (December 20, 2013). http://dx.doi.org/10.15388/lmr.b.2013.28.

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