Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-27T07:34:26.986Z Has data issue: false hasContentIssue false
  • English
  • Français

Triangles whose sides form an arithmetic progression

Published online by Cambridge University Press:  08 October 2020

Sadi Abu-Saymeh  and
Mowaffaq Hajja
Sadi Abu-Saymeh
Affiliation:
2271 Barrowcliffe Dr., Concord, NC 28027, USA e-mail: ssaymeh@yahoo.com
Mowaffaq Hajja
Affiliation:
P. O. Box 388 – Al-Husun, 21510 – Irbid – Jordan e-mail: mowhajja@yahoo.com
Get access Rights & Permissions [Opens in a new window]

Extract

This article is motivated by, and is meant as a supplement to, the recent paper [1]. That paper proves three geometric characterisations of triangles whose sides are in arithmetic progression, or equivalently triangles in which one of the sides is the arithmetic mean of the other two. More precisely, it gives three geometric contexts in which such triangles appear. In this Article, we supply references for the results in [1] and we provide more proofs of these results. We also add more contexts in which such triangles appear, and we raise related issues for future work. We hope that this will be a source of problems for training for, and for including in, mathematical competitions.

Type
Articles
Information
The Mathematical Gazette , Volume 104 , Issue 561 , November 2020 , pp. 469 - 481
Check for updates
Copyright
© Mathematical Association 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Mikami, K., O'Hara, J., and Sugawara, K., Triangles with sides in arithmetic progressions, Elem. Math., 72 (2017) pp. 75–79.Google Scholar
Kimberling, C., Triangle centers and central triangles, Congressus Numerantium 129 (1998) pp. 1–295.Google Scholar
Kimberling, C., Encyclopedia of Triangle Centers, available at https://faculty.evansville.edu/ck6/encyplopedia/ETC.htmlGoogle Scholar
Butchart, J. H., Problem E411, Amer. Math. Monthly, 47 (1940) p. 175; solution by D. L. McKay, ibid., pp. 708–709.Google Scholar
Honsberger, R., Mathematical morsels, Dolciani Math. Expositions, No. 3, Mathematical Association of America, Washington, D. C., 1978.CrossRefGoogle Scholar
Miculita, M. and Brânzei, D., Analogii Triunghi-Tetraedru, Editura Paralela 58, România (2000).Google Scholar
Luenberger, F., Problem E2002, Amer. Math. Monthly, 74 (1967) pp. 720, 860; solution by J. Steinig, ibid. 75 (1968) pp. 909–910.Google Scholar
Chao, W. W., Problem 1506, Math. Mag., 69 (1996), p. 304; solution I by Con Amore Problem Group; solution II by Can A. Minh, ibid. 70 (1997) pp. 302–303.Google Scholar
Hajja, M., Triangle centres: some questions in Euclidean geometry, Int. J. Math. Educ. Sci. Technol. 32 (2001), no. 1, pp. 21–36.Google Scholar
Sastry, K. R. S., Analogies are interesting!, Elem. Math., 59 (2004) pp. 29–36.Google Scholar
Di Pasquale, A., Do, N., and Mathews, D., Problem Solving Tactics, AMT Publishers, Australian Mathematics Trust, University of Canberra, Australia (2014).Google Scholar
Hajja, M. and Spirova, M., A new line associated with the triangle, Elem. Math., 63 (2008) pp. 165–172.CrossRefGoogle Scholar
Andreescu, T. and Feng, Z. (editors), USA and International Mathematical Olympiads 2002, MAA, Washington, D.C. (2003).Google Scholar
Mitchell, B. E., Sextant and bi-sextant triangles, Math. News Letter, 5 (Sept. 1930) pp. 19–24.CrossRefGoogle Scholar
Shirali, S. A., Triangles with one angle equal to 60 degrees, Math. Gaz., 95 (July 2011) pp. 227–234.CrossRefGoogle Scholar