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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Section II - An Overview of the Paper

Summary for the Section :

• In this Section , we shall present an overview of this rather lengthy paper ,
  • briefly stating what each Part of the paper intends to accomplish .

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Part II :

• In general , the centroid of { Triangle A-B-C } usually identified as [ Point O ] ,
  • can be used as the main Point-Of-Reference so that :
    • the position vectors [ Vector O-A ] / [ Vector O-B ] / [ Vector O-C ] will fully define th Triangle .

  We do propose alternately using the center-of-mass for a { 3-equal-mass system } as the main Point-Of-Refernce :

  • when [ Point A ] / [ Point B ] / [ Point C ] happens to lie on the same straight-line .

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Part III :

• Part III is a follow-on on the concepts first presented the author's paper in 2005 :
  • An Approach to the Triangle
    • on how a triangle's size-and-shape may be defined by the 3 key variables ,
      • namely : [ R ] / [ Rho ] / [ Theta ] .

  This part III of the paper then demonstrates :

  • How once the shape-and-size of { Triangle A-B-C } is defined by the variables [ R ] / [ Rho ] / [ Theta ] ,
    • the 2 mirror-imaged Equilateral Triangles will follow-suit from the same [ R ] / [ Rho ] / [ Theta ] .

• The reader may skip over Part III entirely as this is a rather long and tedious part of the paper ,
  • but at the same time , this Part III can be of great importance when doing variation / stability analysis in future .

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Part IV :

• Part IV is the main thrust of the paper and demonstrates :
  • how to solve for the 2 mirror-imaged Equilateral Triangle once the random { Triangle A-B-C } is established ,
    • usually via solving a { 3rd-Order Polynomial Equation } .

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Part V :

• Part V demonstrates that if we did use a different set of { X-Y Axis } on the { plane of the triangle } ,
  • the 2 resultant mirror-imaged Equilateral Triangles would remain the same as in Part IV ,
    • as long as both the old and the new sets of { X-Y Axis } use the same point as the [ Point-of-Origin ]
      • namely [ Point O ] being the centroid of { Triangle A-B-C } .

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Part VI :

• Part VI demonstrates that the 2 mirror-imaged Equilateral Triangles arising from { Triangle A-B-C } :
  • also do not change arising from resizing , i.e. :
    • using 2 differently sized { Unit Circles } old-vs-new to calibrate the distances .

  The 2 mirror-imaged Equilateral Triangles here in Part VI will remain the same as in Part IV .

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Part VII :

• Part VII identifies an Invariant Line for each-and-every triangle other than the Equilateral Triangle ,
  • which will be named the { FENG-Line } for the time being for our discussion purposes.

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

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go to the next Section : Section III - The Centroid as the Center of the Triangle

go to the last Section : Section I - Introduction

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07