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On the largest outscribed equilateral triangle. (English) Zbl 1383.51016

Summary: Given two triangles \(\Delta ABC\) and \(\Delta DEF\), if each side of \(\Delta DEF\) contains a vertex of \(\Delta ABC\), then we call \(\Delta DEF\) an outscribed triangle of \(\Delta ABC\). Given \(\Delta ABC\), let \(\Phi_{\Delta ABC}\) be the set of all outscribed equilateral triangles of \(\Delta ABC\). Clearly \(\Phi_{\Delta ABC}\) is non-empty. In the following we will determine the area of the largest member of \(\Phi_{\Delta ABC}\) when each angle of \(\Delta ABC\) is smaller than \(120^{\circ}\) and show that this largest member can be constructed by ruler and compass from \(\Delta ABC\). The corresponding problem on quadrilaterals has been considered in [D. Zhao, Int. J. Math. Educ. Sci. Technol. 42, No. 4, 534–540 (2011; Zbl 1273.97025)].

MSC:

51M04 Elementary problems in Euclidean geometries
51M25 Length, area and volume in real or complex geometry

Citations:

Zbl 1273.97025
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Full Text: DOI

References:

[1] 1.ZhaoD., Maximal outboxes of quadrilaterals, Int. J. Math. Educ. Sci. Technol. 42 (2011) pp. 534-540.10.1080/0020739X.2010.543166 · Zbl 1273.97025
[2] 2.JohnsonR. A., Modern geometry, An elementary treatise on the geometry of the triangle and the circle, Houghton Mifflin, Boston, MA (1929) pp. 221-222. · JFM 55.0979.01
[3] 3.KimberlingC., Triangle centers and central triangles, Congr. Numer.129 (1998) pp. 1-295. · Zbl 0912.51009
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