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 XLVI - Solving for [ Tau ] and [ Mu ]

Summary for the Section :

• In this Section , we shall solve for [ Tau ] and [ Mu ] , via the equation :
  • 

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Setting up [ OMEGA-1 ] :

• Let us now set up [ OMEGA-1 ] in the usual complex number format :
  • 

    via setting up :

    • the fixed real-value [ R-sub-OMEGA-1 ]

      and

    • the fixed-angle [ Angle THETA-sub-OMEGA-1 ] .

  We then set up the complex fixed-value [ R-sub-Tau-123 ] :

  • 

    being the cube-root of [ R-sub-OMEGA-1 ] .

  We can then set up [ Tau-1 ] / [ Tau-2 ] / [ Tau-3 ] as follows :

  • 

  • 

  • 

    with :

    • 

    • 

    • 

  Thus , we have therefore established the relation :

  • 

• Let us recall { NEW Condition ONE } from Section XLIV ,
  • i.e. this relation :
    • 

    yielding :

    • 

  Consequently , we can now establish the fixed real-value [ R-sub-Mu-123 ] :

  • 

    and also these relations for our use immediately below :

    • 

    • 

    • 

      arising from this relation established back in Section XXII :

      • 

  And based on the above , we can now write the corresponding [ mu ]'s in these usual complex-number formats :

  • 

  • 

  • 

    with :

    • 

    • 

    • 

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Establishing [ X1 ] / [ X2 ] / [ X3 ] :

• Let us recall this relation from Section XLIV :
  • 

    and also :

  • 

  • 

  Thus , we can now write these relations :

  • 

  • 

  • 

  And we then have these final relations for [ X1 ] / [ X2 ] / [ X3 ] :

  • 

  • 

  • 

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What's next :

• With this in hand , we shall take a closer look at the { FENG-Line } , next .

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go to the next Section : Section XLVII - Explaining the { FENG-Line } and the { FENG-Circle }

go to the last Section : Section XLV - Setting up { OMEGA-1 } and { OMEGA-2 }

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07