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On the largest outscribed equilateral triangle

Published online by Cambridge University Press:  23 January 2015

Fengming Dong ,
Dongsheng Zhao  and
Weng Kin Ho
Fengming Dong
Affiliation:
Mathematics and Mathematics Education, National Institute oj Education, Nanyang Technological University, 1 Nanyang Walk, Singapore637616 e-mails:fengming.dong@nie.edu.sg; dongsheng.zhao@nie.edu.sg; wengkin.ho@nie.edu.sg
Dongsheng Zhao
Affiliation:
Mathematics and Mathematics Education, National Institute oj Education, Nanyang Technological University, 1 Nanyang Walk, Singapore637616 e-mails:fengming.dong@nie.edu.sg; dongsheng.zhao@nie.edu.sg; wengkin.ho@nie.edu.sg
Weng Kin Ho
Affiliation:
Mathematics and Mathematics Education, National Institute oj Education, Nanyang Technological University, 1 Nanyang Walk, Singapore637616 e-mails:fengming.dong@nie.edu.sg; dongsheng.zhao@nie.edu.sg; wengkin.ho@nie.edu.sg
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Extract

Given two triangles ΔABC and ΔDEF, if each side of ΔDEF contains a vertex of ΔABC, then we call ΔDEF an outscribed triangle of ΔABC. Given ΔABC, let ΦΔABC be the set of all outscribed equilateral triangles of ΔABC. Clearly ΦΔABC is non-empty. In the following we will determine the area of the largest member of ΦΔABC when each angle of ΔABC is smaller than 120° and show that this largest member can be constructed by ruler and compass from ΔABC. The corresponding problem on quadrilaterals has been considered in [1].

Type
Articles
Information
The Mathematical Gazette , Volume 98 , Issue 541 , March 2014 , pp. 79 - 84
Copyright
Copyright © The Mathematical Association 2014

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References

1. Zhao, D., Maximal outboxes of quadrilaterals, Int. J. Math. Educ. Sci. Technol. 42 (2011) pp. 534–540.CrossRefGoogle Scholar
2. Johnson, R. A., Modern geometry, An elementary treatise on the geometry of the triangle and the circle, Houghton Mifflin, Boston, MA (1929) pp. 221–222.Google Scholar
3. Kimberling, C., Triangle centers and central triangles, Congr. Numer. 129 (1998) pp. 1–295.Google Scholar