Abstract
In a recent paper by Jian Liu (J Math Inequal 8(1), 91–106, 2014), the author establishes a one-parameter family of inequalities pertaining to the distances of a point to the vertices and to the sides of a triangle. His proof depends extremely heavily on algebra packages. In this paper, we give two different simpler and shorter proofs of these inequalities. One of these proofs depends very lightly on computer, and another is a bit longer but all the computations therein are made by hand.
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References
Abu-Saymeh S., Hajja M.: Coincidence of centers for scalene triangles. Forum Geom. 7, 137–146 (2007)
Bottema, O.: Topics in Elementary Geometry, 2nd edn. Springer, New York (2008)
Hajja M.: A generalized Cauchy–Schwarz inequality. Math. Inequal. Appl. 18, 893–899 (2015)
Kimberling C.: Triangle centers as functions. Rocky Mt. J. Math. 23, 1269–1286 (1993)
Kimberling C.: Central points and central lines in the plane of a triangle. Math. Mag. 67, 163–187 (1994)
Kimberling C.: Triangle centers and central triangles. Congr. Numer. 129, 1–285 (1998)
Kimberling C., Moses P.: Inequalities involving six numbers, with applications to triangle geometry. Math. Mag. 85, 221–227 (2012)
Liu J.: A geometric inequality with one parameter for a point in the plane of a triangle. J. Math. Inequal. 8(1), 91–106 (2014)
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This work was supported by a research grant from Yarmouk University.
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Hajja, M., ShahAli, H.A. Simpler proofs of Jian Liu’s inequalities. J. Geom. 108, 565–574 (2017). https://doi.org/10.1007/s00022-016-0359-6
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DOI: https://doi.org/10.1007/s00022-016-0359-6