On Breaking-down any Triangle into
a pair of Mirror-Imaged Equilateral Triangles

by Frank C. Fung ( 1st published in August, 2021. )

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Section XLV - Setting up [ OMEGA-1 ] and [ OMEGA-2 ]

Summary for the Section :

• In this Section , we shall further proceed to solve for the 2 Equilateral Triangles via :
  • setting up [ OMEGA-1 ] and [ OMEGA-2 ] .

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Recalling the 2 NEW Conditons :

• Let us bring in again from the last Section - Section XXXXIV :
  • { NEW Condition ONE } :
    • 

  • { NEW Condition TWO } :
    • 

  And based on { NEW Condition ONE } , we have :

  • 

  Cubing both side the yields us this relation :

  • 

• And on substituting this into { NEW Condition TWO } , we have :
  • 

  Multiplying throughout by [ Tau-cube ] then yields us this equation :

  • 

  And on re-arranging terms , we have :

  • 

  Adding [ H-square ] to both sides then yields us this equation :

  • 

  Consequently , we have :

  • 

  Taking the square-root on both sides then yields us this equation :

  • 

  Consequently , we have :

  • 

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Setting up [ OMEGA-1 ] and [ OMEGA-2 ] :

• Let us now set up fixed complex values [ OMEGA-1 ] and [ OMEGA-2 ] :
  • 

  • 

  And we take note here that :

  • 

• Thus , we can proceed to solve for [ Tau ] and [ Mu ] via :
  • either :
    • 

    or

    • 

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What's next :

• Let us now proceed to solve for the 2 Equilateral Triangles via this equation :
  • 

  next .

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go to the next Section : Section XLVI - Solving for [ Tau ] and [ Mu ]

go to the last Section : Section XLIV - Solving the 3rd-Order Polynomial Equation again in Part VII

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07