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 XLI - Identifying the Invariant { FENG-Line } of the Triangle

Summary for the Section :

• In this Section , we shall identify , at this first instance :
  • the invariant { FENG-Line } of the Triangle .

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Identifying the { FENG-Line } of the Triangle :

• Let us suppose for the moment that :
  • for the random triangle { Triangle A-B-C } :
    • we were able to identify the 2 component mirrored-imaged Equilateral Triangle as :
      • ( Triangle T1-T2-T3 } and { Triangle U1-U2-U3 } ;

      so that :

      • [ 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 below .

  

• We then identify the invariant { FENG-Line } as :
  • the angle-bisector of [ angle T1-O-U1 ] ,

  • the angle-bisector of [ angle T2-O-U2 ] ,

  • the angle-bisector of [ angle T3-O-U3 ] ;

    as per ths set of 3 diagrams below .

  

  

  Let us now take a first look at { FENG-Line } in relation to the 3 triangles ,

  • as per the set of 3 diagrams above .

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What's next :

• In the next 5 Sections to follow , we shall again :
  • set up the { 3rd-Order Polynomial Equation } associated with { Triangle A-B-C } ,

    and

  • solve for the solution mirror-imaged equilateral triangles { Triangle T1-T2-T3 } and { Triangle U1-U2-U3 } ,
    • but with a slightly different twist .

• This then permits us to explain in the Sections thereafter :
  • how the { FENG-Line } would come about ,

    and

  • why the { FENG-Line } is an important structural feature of the Triangle ,
    • especially in the { equal-mass Three-Body Problem } .

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go to the next Section : Section XLII - Recalling the Random Triangle - { Triangle A-B-C }

go to the last Section : Section XL - Outline and Purpose for Part VII

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07