Simpler proofs of Jian Liu's inequalities

摘要

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.