| Part I --- Introduction and Overview |
Section I | Introduction |
Section II | An Overview of the Paper |
| Part II --- Setting up Reference System for the Triangles |
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Section III | The Centroid as the Center of the Triangle |
Section IV | Center-of-Mass for the Three-Body Problem with 3 Equal Masses |
Section V | A First Look at Mirror-Imaged Triangles |
| Part III --- Confirming the Existence of 2 Equilateral Triangles |
Section VI | Outline and Purpose of Part III |
Section VII | Constructing { Triangle A-B-C } |
Section VIII | Some Operation Features of our Construction System |
Section IX | Setting up the 3 Key Variables |
Section X | Analysis for Region One |
Section XI | Analysis for Region Two |
Section XII | Analysis for Region Three |
Section XIII | Analysis for Region Four |
Section XIV | Analysis for Region Five |
Section XV | Analysis for Region Six |
Section XVI | Summarizing our Findings on the 6 Regions |
Section XVII | Confirming the Existance of the 2 Equilateral Triangles |
| Part IV --- Solving for the 2 Equilateral Triangles |
Section XVIII | Outline and Purpose for Part IV |
Section XIX | Setting up the Random Triangle and the Reference Frame |
Section XX | Creating the 3rd-Order Polynomial Equation from { Triangle A-B-C } |
Section XXI | Solving the 3rd-Order Polynomial Equation |
Section XXII | A Quick Review of the Multiplication of Complex Numbers |
Section XXIII | Solving for [ T ] and [ U ] based on [ W1 ] |
Section XXIV | Solving for [ T ] and [ U ] based on [ W2 ] |
Section XXV | Summarizing Our Findings for Part IV |
| Part V --- Invariance of the solution EQ Triangles upon Axis Rotation |
Section XXVI | Outline and Purpose for Part V |
Section XXVII | Setting up the 2 different Reference Systems |
Section XXVIII | Complex Values [ A ] / [ B ] / [ C ] - OLD vs. NEW |
Section XXIX | 3rd-Order Polynomial Equation - OLD vs. NEW |
Section XXX | [ W1 ] and [ W2 ] - OLD vs. NEW |
Section XXXI | [ T1 ] / [ T2 ] / [ T3 ] - OLD vs. NEW |
Section XXXII | [ U1 ] / U2 ] / [ U3 ] - OLD vs. NEW |
Section XXXIII | Summarizing on Part V |
| Part VI --- Invariance of the solution EQ Triangles upon Resizing |
Section XXXIV | Outline and Purpose for Part VI |
Section XXXV | Setting up the Unit Circles - OLD vs. NEW |
Section XXXVI | [ M ] and [ N ] - OLD System vs. NEW System |
Section XXXVII | [ W1 ] and [ W2 ] - OLD System vs. NEW System |
Section XXXVIII | [ T ] / [ U ] / [ X ] - OLD System vs. NEW System |
Section XXXIX | Summarizing on Part VI |
| Part VII --- Identifying the Invariant { FENG-Line } of the Triangle |
Section XL | Outline and Purpose for Part VII |
Section XLI | Identifying the Invariant { FENG-Line } |
Section XLII | Recalling the Random Triangle - { Triangle A-B-C } |
Section XLIII | Recalling the 3rd-Order Polynomial Equation |
Section XLIV | Solving the 3rd-Order Polynomial Equation again in Part VII |
Section XLV | Setting up [ OMEGA-1 ] and [ OMEGA-2 ] |
Section XLVI | Solving for [ Tau ] and [ Mu ] |
Section XLVII | Explaining the { FENG-Line } and the { FENG-Circle } |
Section XLVIII | The { FENG-Line } and { FENG-Circle } in action |
Section XLIX | Principal Axes for a { 3-Equal-Mass System } |
Section L | Confirming the { FENG-Line } as a Principal Axes |
Section LI | Special Comment on the { Equal-Mass Three-Body Problem } |
| Part VIII --- Special Comments and Concluding Remarks |
Section LII | Special Comment on { Mirror-Imaged Regular Pentagons } |
Section LIII | Concluding Remarks |