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 XXXVII - [ W1 ] and [ W2 ] - OLD System vs. NEW System

Summary for the Section :

• In this Section , we shall establish the relation for [ W1 ] and [ W2 ] :
  • under the { OLD System } vs. under the { NEW System } .

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Recalling the Solution Procedure :

• Let us recall from Section XXI that :
  • Once we have reformatted the { 3rd-Order Polynomial Equation } ,
    • 

    we did bring in an Algebriac Identity for comparison purposes :

    • 

  • We then set up the 2 Conditions :
    • Condition ONE :
      • 

    • Condition TWO :
      • 

    and proceeded to solve for a pair of values for [ T ] and [ U ] satisfying the 2 Conditions ,

    • so that the value of [ x ] given by :
      • 

        is a solution value to the reformatted { 3rd-Order Polynomial Equation } .

• And the solution procedure for [ T ] and [ U ] is as follows :
  • First , we brought back { Condition ONE } :
    • 

    And on re-arranging terms , we had :

    • 

    Cubing both sides then yielded us this equation :

    • 

  • We then brought back { Condition TWO } :
    • 

    and substituted therein the result we have immediately above to yield :

    • 

    And on re-arranging terms , we had :

    • 

  • Multiplying throughout by [ T-cube ] then yielded us this equation :
    • 

    And on adding [ N-square ] to both sides of the equation , we had :

    • 

    Consequently :

    • 

  • Taking the square-root on both sides then yielded us this :
    • 

    We then have :

    • 

      or alternately :

    • 

  • We then set up [ W1 ] and [ W2 ] :
    • 

    • 

    so that we would then attempt to solve for [ T ] and [ U ] via 2 different routes , namely :

    • First Route :
      • 

    • Second Route :
      • 

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[ W1 ] and [ W2 ] - OLD vs. NEW :

• Let us now set up again [ W1 ] and [ W2 ] in these formats :
  • Under the { OLD Reference System } :
    • 

    • 

  • Under the { NEW Reference System } :
    • 

    • 

  Let us now recall these results from the last Section - Section XXXVI :

  • 

  • 

  Consequently , we can now write :

  • 

  • 

  Therefore :

  • 

  As a result :

  • 

• Let us bring-in these 2 equations on [ W1 ] for a quick comparison :
  • 

  • 

  And because we have already established above :

  • 

    and

  • 

  it then follows that :

  • 

• Let us also bring-in these 2 equations on [ W2 ] for a quick comparison :
  • 

  • 

  And because we have already established above :

  • 

    and

  • 

  it then follows that :

  • 

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What's next :

• And with this in hand , we shall take a closer look at [ T ] / [ U ] / [ X ] , next .

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go to the next Section : Section XXXVIII - [ T ] / [ U ] / [ X ] - OLD System vs. NEW System

go to the last Section : Section XXXVI - [ M ] and [ N ] - OLD System vs. NEW System

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07