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 XXIV - Solving for [ T ] and [ U ] based on [ W2 ]

Summary for the Section :

• In this Section , we shall solve for the solution values of [ x ] based on this equation :
  • 

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Setting up the Cube-Root of [ W2 ] :

• Let us recall that we did set up [ W2 ] in this format :
  • 

  Let us now also set up the cube-roots of [ W2 ] as being [ T4 ] , [ T5 ] and [ T6 ] respectively .

  We then have :

  • 

  • 

  • 

    where :

    • [ R-sub-T-456 ] is the real-number given by :
      • 

        with :

        • 

      and

    • the angles [ THETA-sub-T4 ] / [ THETA-sub-T5 ] / [ THETA-sub-T6 ] are given by :
      • 

      • 

      • 

  Thus , we have now established this relation :

  • 

• And we take special note at this point that :
  • the complex values [ T4 ] / [ T5 ] / [ T6 ] are in the formation of an equilateral triangle on the Complex Plane ,
    • with the { T4-T5-T6 rotation-order } being a { counter-clockwise rotation-order } .

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Setting up [ U4 ] /[ U5 ] / [ U6 ] :

• Let us recall again { Condition ONE } and { Condition TWO } from Section XXI that brought us here :
  • Condition ONE :
    • 

  • Condition TWO :
    • 

• And consequently , we have :
  • 

  We then set up :

  • [ U4 ] corresponding to [ T4 ] :
    • 

  • [ U5 ] corresponding to [ T5 ] :
    • 

  • [ U6 ] corresponding to [ T6 ] :
    • 

• Let us recall that we did previously set up [ M ] in this format :
  • 

  We then have :

  • 

  • 

  • 

• Let us now bring in the { Special Type of Reciprical Relation } established previously in Section XXII :
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  We can then re-write the above 3 equations in this manner :

  • 

  • 

  • 

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

  We then have :

  • 

  • 

  • 

• And we take special note at this point that :
  • the complex values [ U4 ] / [ U5 ] / [ U6 ] are in the formation of an equilateral triangle on the Complex Plane ,
    • with the { U4-U5-U6 rotation-order } being a { clockwise rotation-order } .

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Setting up the Solution Set [ X4 ] / [ X5 ] / [ X6 ] :

• We can now set up a solution set for the { 3rd-Order Polynomial Equation } :
  • 

  We then have :

  • 

  • 

  • 

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What's next :

• Let us now summarize our findings in this Part IV , next .

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go to the next Section : Section XXV - Summarizing Our Findings for Part IV

go to the last Section : Section XXIII - Solving for [ T ] and [ U ] based on [ W1 ]

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07