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Iterating circum-medial triangles. (English) Zbl 1507.51011

Summary: When considering ‘convergence’ many people think of number sequences or contexts arising from calculus. But there are also interesting phenomena of convergence – easy to visualize – arising in elementary geometry. Some of them are so elementary that they can be dealt with at school, for instance an example of iteration that is described in [D. Ismailescu and J. Jacobs, Period. Math. Hung. 53, No. 1–2, 169–184 (2006; Zbl 1127.52019), p. 171f] (see also [S. Jones, “Two iteration examples”, Math. Gaz. 74, No. 467, 58–62 (1990; doi:10.2307/3618858), p. 59], [S. Abbott, “Averaging sequences and triangles”, Math. Gaz. 80, No. 487, 222–224 (1996; doi:10.2307/3620354), p. 222f] and [M. De Villiers, “Over and over again: two geometric iterations with triangles”, Learning & Teaching Math. 16, 40–45 (2014), p. 42ff]).

MSC:

51M04 Elementary problems in Euclidean geometries

Citations:

Zbl 1127.52019
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Full Text: DOI

References:

[1] D.Ismailescu, J.Jacobs, On sequences of nested triangles, Periodica Mathematica Hungarica, 53 (1-2) (2006) pp. 169-184.10.1007/s10998-006-0030-3 · Zbl 1127.52019
[2] StephenJones, Two iteration examples, Math. Gaz.74 (March 1990) pp. 58-62.10.2307/3618858
[3] S.Abbott, Averaging sequences and triangles, Math. Gaz.80 (March 1996) pp. 222-224.
[4] M.De Villiers, Over and over again: two geometric iterations with triangles, Learning & Teaching Mathematics 16 (July 2014) pp. 40-45, also available at http://dynamicmathematicslearning.com/geometriciteration-examples.pdf
[5] H.Humenberger, F.Embacher, Convergence with respect to triangle shapes – elementary geometric iterations, Learning & Teaching Mathematics25 (December 2018) pp. 15-19.
[6] S. Y.Trimble, The limiting case of triangles formed by angle bisectors, Math. Gaz.80 (November 1996) pp. 554-556.
[7] G.Leversha, The geometry of the triangle, United Kingdom Mathematics Trust (2013). · Zbl 1384.05029
[8] G.Bourgeois, J.-P.Lechêne, Etude d’une itération en géométrie du triangle, Bulletin de l’APMEP 1(409) (1997) pp. 147-154 also available at https://www.apmep.fr/IMG/pdf/AAA97013.pdf
[9] H.Humenberger, F.Embacher, Konvergenz bei Dreiecken – interessante geometrische Iterationen, Mathematische Semesterberichte 65 (2) (2018) pp. 253-275 also available at http://link.springer.com/article/10.1007/s00591-017-0206-3 · Zbl 1419.51011
[10] G.Bourgeois, S.Orange, Dynamical systems of simplices in dimension two or three. In: T. Sturm, Ch. Zengler, (Ed., 2011): Automated deduction in geometry. 7th International Workshop, Shanghai, China, September 22-24, 2008. Springer Berlin-Heidelberg, 1-21 also available at https://arxiv.org/pdf/0906.0352.pdf · Zbl 1302.51014
[11] R.Honsberger, Episodes in nineteenth and twentieth century Euclidean geometry, The Mathematical Association of America (1995). · Zbl 0829.01001
[12] M. F.Gomez, P.Taslakian, G. T.Toussaint, Convergence of the shadow sequence of inscribed polygons, 18th Fall Workshop on Computational Geometry, New York, EEUU (2008). http://ac.aua.am/ptaslakian/web/publications/shadowConvergence.pdf
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