Remarks on a Generalization of a Question Raised by Pál Erdős Concerning a Geometric Inequality in Acute Triangles II

• Main Author: Béla FINTA
• Format: Article
• Language: English
• Published: Editura Universităţii "Petru Maior" 2018-06-01
• Series: Scientific Bulletin of the ''Petru Maior" University of Tîrgu Mureș
• Subjects: geometrical inequalities; acute triangle; inerior bisectrices
• Online Access: http://scientificbulletin.upm.ro/papers/2018-1/7_Remarks%20on%20a%20generalization_Finta.pdf

The purpose of this paper is to give a negative answer to a possible generalization of an open question raised by Pál Erd ˝os, concerning an inequality in acute triangles. We prove here that from a < b < c does not follow a 2k+1 + (la) 2k+1 < b2k+1 + l 2k+1 b < c 2k+1 + l 2k+1 c in every acute triangle ABC, nor the opposite chain of inequalities, where k ∈ N, k ≥ 2, and a, b, c denotes the length of the triangles sites, while la, lb, lc denotes the length of the interior angle bisectors, as usual. We achieve this by constructing effectively two counterexamples, one for each type of inequalities.