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Long medians and long angle bisectors

Published online by Cambridge University Press:  13 October 2021

Sadi Abu-Saymeh ,
Yaqeen Al-Momani ,
Mowaffaq Hajja  and
Mostafa Hayajneh
Sadi Abu-Saymeh
Affiliation:
2271 Barrowcliffe Dr., Concord, NC 28027, USA, e-mail: ssaymeh@yahoo.com
Yaqeen Al-Momani
Affiliation:
Mathematics – Ajloun National University Ajloun – Jordan, e-mail: yaqeenal-momany@yahoo.com
Mowaffaq Hajja
Affiliation:
Mathematics – Philadelphia University Amman – Jordan, e-mail: mowhajja@yahoo.com
Mostafa Hayajneh
Affiliation:
Mathematics – Yarmouk University Jordan, e-mail: hayaj86@yahoo.com
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Extract

Following Euler, we denote the side lengths and angles of a triangle ABC by a, b, c, A, B, C in the standard order. Any line segment joining a vertex of ABC to any point on the opposite side line will be called a cevian, and a cevian AA′ of length t will be called long, strictly long, or balanced according as t ≥ a, t > a or t = a. If A′ lies strictly between B and C, AA′ is called an internal cevian. This convention regarding cevians is not universal, and it is, for example, in a heavy contrast with that in [1, p. 73].

Type
Articles
Information
The Mathematical Gazette , Volume 105 , Issue 564 , November 2021 , pp. 397 - 409
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Copyright
© The Mathematical Association 2021

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References

Leversha, G., The geometry of the triangle, UKMT (2013).Google Scholar
Abu-Saymeh, S., Hajja, M., and Hayajneh, M., Long altitudes and an unexpected appearance of the golden ratio, Math. Gaz. 102 (November 2018) pp. 523-527.CrossRefGoogle Scholar
Yiu, P., Euclidean geometry, preliminary version, Department of Mathematics, Florida Atlantic University (Fall 1998).Google Scholar
Cvetkovski, Z., Inequalities: theorems, techniques and selected problems, Springer (2012).Google Scholar
Hajja, M., Stronger forms of the Steiner-Lehmus theorem, Forum Geom. 8 (2008) pp. 157-161.Google Scholar
Abu-Saymeh, S. and Hajja, M., More variations of the Steiner-Lehmus theme, Math. Gaz. 103 (March 2019) pp. 1-11.CrossRefGoogle Scholar
Altinaş, A., Some collinearities in the heptagonal triangle, Forum Geom. 16 (2016) pp. 249-256.Google Scholar
Yiu, P., Heptagonal triangle and their companies, Forum Geom. 9 (2009) pp. 125-148.Google Scholar
Bankoff, L., and Garfunkel, J., The heptagonal triangle, Math. Mag. 46 (1973) No. 1, pp. 7-19.CrossRefGoogle Scholar
Primrose, E. J. F., An interesting triangle, Math. Gaz. 63 (March1979) pp. 42-43CrossRefGoogle Scholar
Parry, C. F., Steiner-Lehmus and the Feuerbach triangles, Math. Gaz. 79 (July 1995) pp. 275-285.CrossRefGoogle Scholar
Johnson, R. A., Advanced Euclidean geometry, Dover Publications (1929).Google Scholar
Mironescu, P. and Panaitopol, L., The existence of a triangle with prescribed angle bisector lengths, Amer. Math. Monthly 101 (1994) pp. 58-60.CrossRefGoogle Scholar
Dinca, G. and Mawhin, J., A constructive fixed point approach to the existence of a triangle with prescribed angle bisector lengths, Bull. Belg. Math. Soc. Simon Stevin, 17 (2010) pp. 333-341.CrossRefGoogle Scholar
Osinkin, S. F., On the existence of a triangle with prescribed bisector lengths, Forum Geom. 16 (2016) pp. 399-405.Google Scholar
Heindl, G., How to compute a triangle with prescribed lengths of its angle bisectors, Forum Geom. 16 (2016) pp. 407-414.Google Scholar