Triangles whose sides form an arithmetic progression. (English) Zbl 1507.51006
Summary: This article is motivated by, and is meant as a supplement to, the recent paper [K. Mikami et al., Elem. Math. 72, No. 2, 75–79 (2017; Zbl 1372.51011)]. 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 [loc. cit.] 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.
MSC:
| 51M04 | Elementary problems in Euclidean geometries |
Citations:
Zbl 1372.51011References:
| [1] | Mikami, K., O’Hara, J., and Sugawara, K., Triangles with sides in arithmetic progressions, Elem. Math., 72 (2017) pp. 75-79. · Zbl 1372.51011 |
| [2] | Kimberling, C., Triangle centers and central triangles, Congressus Numerantium129 (1998) pp. 1-295. · Zbl 0912.51009 |
| [3] | Kimberling, C., Encyclopedia of Triangle Centers, available at https://faculty.evansville.edu/ck6/encyplopedia/ETC.html · Zbl 0803.51017 |
| [4] | Butchart, J. H., Problem E411, Amer. Math. Monthly, 47 (1940) p. 175; solution by D. L. McKay, ibid., pp. 708-709. |
| [5] | Honsberger, R., Mathematical morsels, Dolciani Math. Expositions, No. 3, Mathematical Association of America, Washington, D. C., 1978. · Zbl 0378.00021 |
| [6] | Miculita, M. and Brânzei, D., Analogii Triunghi-Tetraedru, Editura Paralela 58, România (2000). |
| [7] | Luenberger, F., Problem E2002, Amer. Math. Monthly, 74 (1967) pp. 720, 860; solution by J. Steinig, ibid. 75 (1968) pp. 909-910. |
| [8] | 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. |
| [9] | Hajja, M., Triangle centres: some questions in Euclidean geometry, Int. J. Math. Educ. Sci. Technol. 32 (2001), no. 1, pp. 21-36. · Zbl 1011.51006 |
| [10] | Sastry, K. R. S., Analogies are interesting!, Elem. Math., 59 (2004) pp. 29-36. · Zbl 1055.51005 |
| [11] | Di Pasquale, A., Do, N., and Mathews, D., Problem Solving Tactics, AMT Publishers, Australian Mathematics Trust, University of Canberra, Australia (2014). |
| [12] | Hajja, M. and Spirova, M., A new line associated with the triangle, Elem. Math., 63 (2008) pp. 165-172. · Zbl 1390.51010 |
| [13] | Andreescu, T. and Feng, Z. (editors), USA and International Mathematical Olympiads 2002, MAA, Washington, D.C. (2003). · Zbl 1054.00003 |
| [14] | Mitchell, B. E., Sextant and bi-sextant triangles, Math. News Letter, 5 (Sept. 1930) pp. 19-24. |
| [15] | Shirali, S. A., Triangles with one angle equal to 60 degrees, Math. Gaz., 95 (July 2011) pp. 227-234. · Zbl 1383.51028 |
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