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Inequalities about the area bounded by three cevian lines of a triangle

Yagub N. Aliyev

TL;DR

This paper addresses sharp area inequalities for triangles formed by cevians in a triangle. It introduces a general bound, derived via a discrete Hölder inequality, on the ratio $ rac{ ext{Area}( riangle RST)}{ ext{Area}( riangle DEF)}$ in terms of $\\lambda_1,\\lambda_2,\\lambda_3$ and $u,v,w$, and identifies equality conditions when $\alpha=\beta=\gamma$. The authors show how this general inequality yields corollaries of Schlömilch's and Zetel's concurrency theorems, provides a new refinement of Rigby's inequality, and discusses special cases (e.g., $uvw=1$) with connections to Möbius-type area relations. They also pose an open problem for the $uvw=1$ case and emphasize the method's potential to prove concurrency results via area considerations.

Abstract

In the paper we prove generalization of Schlömilch's and Zetel's theorems about concurrent lines in a triangle. This generalization is obtained as a corollary of sharp geometric inequality about the ratio of triangular areas which is proved using discrete variant of Hölder's inequality. Also a new sharp refinement of J.F. Rigby's inequality, which itself generalized Möbius theorem about the areas of triangles formed by cevians of a triangle, is proved.

Inequalities about the area bounded by three cevian lines of a triangle

TL;DR

This paper addresses sharp area inequalities for triangles formed by cevians in a triangle. It introduces a general bound, derived via a discrete Hölder inequality, on the ratio in terms of and , and identifies equality conditions when . The authors show how this general inequality yields corollaries of Schlömilch's and Zetel's concurrency theorems, provides a new refinement of Rigby's inequality, and discusses special cases (e.g., ) with connections to Möbius-type area relations. They also pose an open problem for the case and emphasize the method's potential to prove concurrency results via area considerations.

Abstract

In the paper we prove generalization of Schlömilch's and Zetel's theorems about concurrent lines in a triangle. This generalization is obtained as a corollary of sharp geometric inequality about the ratio of triangular areas which is proved using discrete variant of Hölder's inequality. Also a new sharp refinement of J.F. Rigby's inequality, which itself generalized Möbius theorem about the areas of triangles formed by cevians of a triangle, is proved.
Paper Structure (4 sections, 4 theorems, 29 equations, 6 figures)

This paper contains 4 sections, 4 theorems, 29 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Conclusion
  4. Declarations

Key Result

Theorem 2.1

Let $D$ and $K$, $E$ and $L$, $F$ and $M$ be arbitrary points on sides $BC$, $AC$, and $AB$, repsectively, of a triangle $ABC$. Denote $AK\cap EF=N$, $BL\cap DF=Q$, $CM\cap DE=P$, $DN\cap EQ=R$, $FP\cap EQ=S$, and $FP\cap DN=T$. Denote also $\frac{|BD|}{|DC|}=\lambda_1$, $\frac{|CE|}{|EA|}=\lambda_2

Figures (6)

  • Figure 1: Steiner-Routh's theorem.
  • Figure 2: Zetel's generalization of Schlömilch's theorem.
  • Figure 3: Inequality about the ratio of areas of $\triangle RST$ and $\triangle DEF$.
  • Figure 4: Generalization of Zetel's theorem.
  • Figure 5: A new proof of Zetel's generalization of Schlömilch's theorem.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3
  • proof
  • Theorem 2.4
  • proof