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 XXIX - 3rd-Order Polynomial Equation - OLD vs. NEW

Summary for the Section :

• In this Section , we shall make a quick comparison of the { 3rd-Order Polynomial Equation } :
  • under the { OLD Reference System } vs. the { NEW Reference System } .

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Expanding on the previous 3 Equations :

• Let us now bring back these 3 equations established in the last Section-Section XXVII :
  • 

  • 

  • 

• Consequently , we can now write :
  • 

  • 

  • 

  And we also have :

  • 

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Bringing in the { 3rd-Order Polynomial Equation } :

• Let us recall from Section XXI that :
  • In order to solve for the 2 Mirror-Imaged Equilateral Triangles ,
    • we did assume that [ A ] / [ B ] / [ C ] are the solution values for a { 3rd-Order Polynomial Equation } of the format :
      • 

      so that these 2 equations below are to be deemed equivalent throughout :

      • 

      • 

  • And on expanding the 2nd equation above , we did establish that :
    • 

    • 

    • 

• And because [ Point O ] is the centroid of { Triangle A-B-C } , we always have :
  • [ Vector O-A ] + [ Vector O-B ] + [ Vector O-C ] = [ zero ]

  so that :

  • the complex values [ A ] , [ B ] and [ C ] always add up to [ zero ] , i.e. :
    • 

  Consequently :

  • 

• Thus , we can re-write the original equation :
  • 

  as :

  • 

  And on re-arranging terms , we have this reformatted { 3rd-Order Polynomial Equation } :

  • 

• Thus , we can now set up these 2 equations for our comparison purposes :
  • Under the { OLD Reference System } :
    • 

  • Under the { NEW Reference System } :
    • 

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[ K1 ] and [ K0 ] - NEW vs. OLD :

• We then set up :
  • 

    and

  • 

  Let us now bring back these 3 equations from above :

  • 

  • 

  • 

  Consequently , we have :

  • 

• Let us also set up :
  • 

    and

  • 

  Let us bring back from above this equation :

  • 

  Consequently, we can now write :

  • 

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[ M ] and [ N ] - OLD vs. NEW :

• Let us recall from Section XXI that :
  • We did convert the reformatted { 3rd-Order Polynomial Equation } from above , i.e. this equation :
    • 

    into this new format :

    • 

    via setting up [ M ] and [ N ] as follows :

    • 

      and

    • 

• Let us now set up these 2 equations for comparison purposes :
  • Under the { OLD Reference System } :
    • 

      with :

      • 

      • 

  • Under the { NEW Reference System } :
    • 

      with :

      • 

      • 

• Since we have already established above that :
  • 

    and

  • 

  it then automatically follows that :

  • 

    and

  • 

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What's Next :

• And with this in hand , we are now ready take a quick look at [ W1 ] and [ W2 ] ,
  • under the { OLD Reference Syetem } vs. the { NEW Reference System } ;

  next .

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go to the next Section : Section XXX - [ W1 ] and [ W2 ] - OLD vs. NEW

go to the last Section : Section XXVIII - [ A ] / [ B ] / [ C ] - OLD vs. NEW

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07