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A New Lemoine-Type Circle

Miłosz Płatek

Abstract

This paper presents a new Lemoine-type circle defined by a six-point configuration satisfying a cocyclicity criterion. We prove the result, establish a converse theorem, and relate the new circle to previously known Lemoine circles, in particular the one introduced by Q.T. Bui. We show that the new circle does not belong to the family of Tucker circles.

A New Lemoine-Type Circle

Abstract

This paper presents a new Lemoine-type circle defined by a six-point configuration satisfying a cocyclicity criterion. We prove the result, establish a converse theorem, and relate the new circle to previously known Lemoine circles, in particular the one introduced by Q.T. Bui. We show that the new circle does not belong to the family of Tucker circles.

Paper Structure

This paper contains 6 sections, 8 theorems, 6 equations, 9 figures.

Table of Contents

  1. Historical Background
  2. The Family of Tucker Circles
  3. Known Lemoine Circles
  4. New Lemoine-type Circle
  5. Converse Theorem
  6. Lemoine Circle Centers Along the Brocard Axis

Key Result

Proposition 2.2

The vertices of the Tucker hexagon lie on a single circle, called the Tucker circle, whose center lies on the line $OL$. $\blacktriangleleft$$\blacktriangleleft$

Figures (9)

  • Figure 1: First Lemoine Circle
  • Figure 2: Second Lemoine Circle
  • Figure 3: Third Lemoine Circle
  • Figure 4: Q.T.Bui Circle
  • Figure 5: Theorem \ref{['new']}.
  • ...and 4 more figures

Theorems & Definitions (13)

  • Definition 2.1: Tucker circle
  • Proposition 2.2
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Remark 3.5
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • ...and 3 more