On Breaking-down any Triangle into
a pair of Mirror-Imaged Equilateral Triangles

by Frank C. Fung ( 1st published in September, 2021. )

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Summary and Key Findings :

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SUMMARY :

• This paper is an attempt to break-down any { 3-Points Shape } , such as a Triangle ,
  • into a pair of mirror-imaged Equilateral Triangle , resized and/or re-oriented as the case may be .

  
  

• We then have the relations :
  • [ Vector O-A ] = [ Vector O-T1 ] + [ Vector O-U1 ]

  • [ Vector O-B ] = [ Vector O-T2 ] + [ Vector O-U2 ]

  • [ Vector O-C ] = [ Vector O-T3 ] + [ Vector O-U3 ]

    as per the set of 3 diagrams above .

  In general , the centroid of the Triangle is used as the primary Point-of-Reference ,

  • where as in the case of syzygy ( Point A/B/C lying on the same straight-line ) :
    • the center-of-mass for a { 3-equal-mass system } of [ Mass A ] / [ Mass B ] / [ Mass C ]
      • is used instead as the primary Point-of-Reference .

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Key Finding One :

• We were successful in breaking-down any { 3-Points Shape } into its 2 mirror-imaged Equilateral Triangles .

  This done via :

  • First creating a random complex plane in the plane of the triangle , so that :
    • the { REAL Axis } and the { IMAGINARY Axis } intersect at [ Point O ] , the centroid of { Triangle A-B-C } ;

  • Secondly assign complex values to Point A / B / C based on their position on the Complex Plane ,

  • Thirdly , asssume that the complex values A / B / C are the solution values to a { 3rd-Order Polynomial Equation } of the format :
    • 

      and proceed to calculate the fixed values K2 / K1 / K0 based on the complex values A / B / C .

  • Fourthly , solve the { 3rd-Order Polynomial Equation } via the Cardano Method ,
    • and this will yield us the 2 Equilateral Triangles .

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Key Finding Two :

• We were able to establish that the 3 Triangles , namely :
  • the original { Triangle A-B-C } and its 2 broken-down Equilateral Triangles ,

    are invariant upon Axis-Rotation and Distance Re-calibration ,

    • in the solution procedure as detailed in { Key Finding One } above .

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Key Finding Three :

• We identified an invariant-line associated with each-and-every triangle other than the Equilateral Triangle ,
  • which we have named the { FENG-Line } for the time being to facilitate further discussions .

  The { FENG-Line } also happens to be a Principal Axes of Rotation for a ( 3-equal-mass system } .

• The problem with the Equilateral Triangle here is two-folds :
  • When we break-down an Equilateral Triangle into 2 component mirror-imaged Equilateral Triangles :
    • one of the broken-down Equilateral Triangle is the same size as the original Equilateral Triangle ,
      • whereas the other broken-down Equilateral Triangle is of size [ zero ] .

  • The original Equilateral Triangle will have multiple Principal Axes of Rotation .

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Key Finding Four :

• We do observe here that many repetitive orbits for the { Equal-Mass Three-Body Problem } have been found .

  We hope that :

  • via breaking-down any { 3-Points Shape } into its 2 component mirror-imaged Equilateral Triangles ,

    this will lead to a better understanding of the structural features and the stability of these orbits .

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Table of Content

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go to the 1st Section : Section I - Introduction

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Original dated 2021-9-07