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 XXVIII - Complex Values [ A ] / [ B ] / [ C ] - OLD vs. NEW

Summary for the Section :

• In this Section , we shall set up the equations relating :
  • the Complex Values [ A ] / [ B ] / [ C ] under the { NEW Reference System }

    and

  • the Complex Values [ A ] / [ B ] / [ C ] under the { OLD Reference System } .

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{ NEW Reference System } vs. { OLD Reference System } :

• Let us recall that we did set up , in the last Section - Section XXVII ,
  • the complex values of [ A ] / [ B ] / [ C ] under the { NEW Reference System } as :
    • 

    • 

    • 

  And we did also set up the relations between the { OLD angles } and the { NEW angles }in that Section :

  • 

  • 

  • 

  Consequently , we can now re-write the first 3 equations above as :

  • 

  • 

  • 

• Let us now bring in this { Algebraic Identity on Complex Multiplication } previously established in Section XXII :
  • 

  Applying this to the 3 equations above then yields us these 3 new equations below :

  • 

  • 

  • 

• Let us also recall that the complex values of [ A ] / [ B ] / [ C ] under the { OLD Reference System } are :
  • 

  • 

  • 

  Substituting these into the previous 3 equations above then yields us these 3 new equations :

  • 

  • 

  • 

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What's Next :

• With this in hand , we are then ready to look at the { 3rd-Order Polynomial Equation } :
  • 

    for a quick comparison under the { OLD Reference System } and the { NEW Reference System } ;

  next .

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go to the next Section : Section XXIX - 3rd-Order Polynomial Equation - OLD vs. NEW

go to the last Section : Section XXVII - Setting up the 2 different Reference Systems

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07