Abstract
We establish a new generalization of the Erdös–Mordell inequality by adding more a set of weights to its terms. The same method is used on two other variants of the Erdös–Mordell inequality which are Barrow’s inequality and Dao–Nguyen–Pham’s inequality. Using these generalizations, we derived some strengthened versions of the original Erdös–Mordell inequality and its variations.
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References
Erdös, P.: Problem 3740. Am. Math. Mon. 42, 396 (1935)
Mordell, L.J., Barrow, D.F.: Solution 3740. Am. Math. Mon. 44, 252–254 (1937)
Dar, S., Gueron, S.: A weighted Erdös–Mordell inequality. Am. Math. Mon. 108, 165–168 (2001)
Oppenheim, A.: The Erdös–Mordell inequality and other inequalities for a triangle. Am. Math. Mon. 68, 226–230 (1961)
Oppenheim, A.: New inequalities for a triangle and an internal point. Ann. Mat. Pura Appl. 53, 157–163 (1961)
Pech, P.: Erdös–Mordell inequality for space n-gons. Math. Pannon. 5, 3–6 (1994)
Alsina, C., Nelsen, R.B.: A visual proof of the Erdös–Mordell inequality. Forum Geom. 7, 99–102 (2007)
Dao, T.O., Nguyen, T.D., Pham, N.M.: A strengthened version of the Erdös–Mordell inequality. Forum Geom. 16, 317–321 (2016)
Liu, J.: A Geometric inequality with one parameter for a point in the plane of a triangle. J. Math. Inequal. 1, 91–106 (2014)
Liu, J.: Refinements of the Erdös–Mordell inequality, Barrow’s inequality, and Oppenheim’s inequality. J. Inequal. Appl. 9, 1–18 (2016)
Liu, J.: New refinements of the Erdös–Mordell inequality. J. Math. Inequal. 12, 63–75 (2018)
Liu, J.: Two new weighted Erdös–Mordell type inequalities. Discrete Comput. Geom. 59, 707–724 (2018)
Liu, J.: An improved result of a weighted trigonometric inequality in acute triangles with applications. J. Math. Inequal. 14, 147–160 (2020)
Liu, J.: Sharpened versions of the Erdös–Mordell inequality. J. Inequal. Appl. 206, 1–12 (2015)
Wolstenholme, J.: A Book of Mathematical Problems. Macmillan, Cambridge, London (1867)
Malešević, B., Petrović, M.: Barrow’s inequality and signed angle bisectors. J. Math. Inequal. 8, 537–544 (2014)
Acknowledgements
The author wishes to thank Professor Floor van Lamoen from the Netherlands for his help in reading this manuscript and making important comments. The author would like to thank Professor Hans Havlicek and Professor Kreuzer for their encouragement to me throughout this manuscript. The author is grateful to the referee for the careful reading of the paper and valuable comments. The author also wishes to express appreciation to Professor Bonnie Ponce from the Mathematical Association of America for her encouragement from the beginning of this manuscript.
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Tran, Q.H. A family of weighted Erdös–Mordell inequality and applications. J. Geom. 112, 33 (2021). https://doi.org/10.1007/s00022-021-00597-0
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DOI: https://doi.org/10.1007/s00022-021-00597-0