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Proof of the Generalization of the Sawayama-Thébault Theorem

Miłosz Płatek

Abstract

We prove two conjectures posed in 2016 concerning a generalization of the Sawayama-Thébault Theorem and the Sawayama Lemma. We show that this generalized statement can be viewed in Laguerre geometry, which provides a natural framework for resolving the problem.

Proof of the Generalization of the Sawayama-Thébault Theorem

Abstract

We prove two conjectures posed in 2016 concerning a generalization of the Sawayama-Thébault Theorem and the Sawayama Lemma. We show that this generalized statement can be viewed in Laguerre geometry, which provides a natural framework for resolving the problem.
Paper Structure (3 sections, 8 theorems, 24 equations, 17 figures)

This paper contains 3 sections, 8 theorems, 24 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Generalization of the Sawayama Lemma
  3. Generalization of the Sawayama--Thébault Theorem

Key Result

Proposition 1.1

Let $ABC$ be a triangle, and let $D$ be a point on the side $BC$ of triangle $ABC$. Let $O_1$ and $O_2$ ($O_1 \neq O_2$) be the centers of circles tangent to segment $BC$, to segment $AD$, and to the circumcircle of triangle $ABC$. Then, regardless of the choice of point $D$, the line $O_1O_2$ passe

Figures (17)

  • Figure 1: The Sawayama--Thébault Theorem.
  • Figure 2: The Sawayama Lemma.
  • Figure 3: The Sawayama Lemma in an alternative configuration.
  • Figure 4: $A'I$ is the angle bisector of angle $BA'C$.
  • Figure 5: Cyclicity of quadrilaterals $A'C_1IB$ and $AB_1IC$.
  • ...and 12 more figures

Theorems & Definitions (15)

  • Proposition 1.1: The Sawayama--Thébault Theorem
  • Proposition 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3: Generalized Sawayama Lemma
  • proof
  • Lemma 2.4
  • proof
  • ...and 5 more