On the extension of the Erdös-Mordell type inequalities (original) (raw)

Several Geometric Inequalities of Erdös - Mordell Type in the Convex Polygon

2012

In this paper we present the several geometric inequalities of Erdos-Mordell type in the convex polygon, using the Cauchy Inequality. In (6), in colaboration with A. Gobej, we present some geometric inequal- ities of Erdos-Mordell type in the convex polygon. Here, we found others geometric inequalities of Erdos-Mordell type, using several known inequali- ties, in the convex polygon. Let A1,A2;...,An the vertices of the convex polygon, n � 3; and M, a point interior to the polygon. We note with Rk the distances from M to the vertices Ak and we note with rk the distances from M to the sides (AkAk+1) of length AkAk+1 = ak; where k = 1;n and An+1 � A1. For all k 2 f1;:::;ng with An+1 � A1 and m � \

Erdos-Mordell Inequality in Taxicab Geometry

2019

In this paper is considered Erdos-Mordell's inequality for the triangle triangleABC\triangle ABCtriangleABC in the Taxicab plane geometry.$\;\;$It is shown that for the Erdos-Mordell's inequality RA+RB+RC,geq,w,(ra+rb+rc)R_A + R_B + R_C \, \geq \, w \, (r_a + r_b + r_c)RA+RB+RC,geq,w,(ra+rb+rc) holds for triangles in appropriate positions, if w=3/2w = 3/2w=3/2.

Generalising a triangle inequality

The Mathematical Gazette, 2018

The main goal of this paper is to give a deeper understanding of the geometrical inequality proposed by Martin Lukarevski in [1]. In order to formulate our results we shall introduce and use the following notation throughout this paper. Let A1A2A3 be a triangle a1, a2, a3, the lengths of the sides opposite to A1, A2, A3 respectively, P an arbitrary inner point of it xi, the distance of P from the side of length ai. Let r, R be the inradius and circumradius of the triangle hi, the altitude belonging to side ai, Δ the area and finally let α be a real parameter. We adopt also the use of Σui to refer the sum taken over the suffices i = 1, 2, 3. Now we are in the position to reformulate the original problem into a more general form namely: find bounds for Σxαi in terms of r and R. The main results of our investigation are summarised in the following theorem.

A PROOF OF TWO CONJECTURES RELATED TO THE ERDÖS-DEBRUNNER INEQUALITY

In this paper we prove some results which imply two conjectures proposed by Janous on an extension to the p-th power-mean of the Erdös-Debrunner inequality relating the areas of the four sub-triangles formed by connecting three arbitrary points on the sides of a given triangle.

New Proofs of Triangle Inequalities

arXiv: General Mathematics, 2018

In this article, we give three new proofs of the triangle inequality in Euclidean Geometry. There seems to be only one known proof at the moment. It is due to properties of triangles, but our proofs are due to circles or ellipses. We aim to prove the triangle inequality as simple as possible without using properties of triangles.

On the inequalities of Erdös-Turán and Berry-Esseen, II

Proceedings of the Japan Academy. Series A, Mathematical sciences, 1989

This is continued rom [1]. 5. The ideas of the proofs of the results given in Sections 3 and 4 are similar. Here we shall prove only Theorem 1. The proof is based on some ideas o Sendov [3] and the author [2]. We begin with a well known lemma of Sendov, which he used in the approximation theory.