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 XXII - A Quick Review of the Multiplication of Complex Numbers

Summary for the Section :

• In this Section , we shall do a quick review on the { multipliction of Complex Numbers } ,
  • and further set up 2 more { complex number relations } for our use here in Part IV
    • and later in Part V / VI / VII of this paper .

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The Product of 2 complex numbers [ P ] and [ Q ] :

• Let [ P ] and [ Q ] be two (2) non-zero complex numbers .

  

  We then set up :

  • 

  • 

    where :

    • [ R-sub-P ] and [ R-sub-Q ] are Real Numbers greater than [ zero ] , i.e. :
      • 

        and

      • 

  Let us now further expand on [ P ] and [ Q ] such that :

  • 

  • 

• The product of [ P ] and [ Q ] is then :
  • 
    

  And on consolidating terms , we have :

  • 
    

  Noting here that :

  • 

  we can now re-write the above equation as follows :

  • 
    

• Consequently, we have this final format for the product of [ P ] and [ Q ] :
  • 

    arising from :

    • 
    • 

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An Algebraic Identity on Complex Multiplication :

• Let us also develop here an { Algebraic Identity on Complex Multiplication } for our later use .

  We then bring in this equation involving the general angles [ PHI ] and [ PSI ] :

  • 

    

  We take note here again that :

  • 

  Consequently , we can re-write the above equation as follows :

  • 

• And on switching sides , we have :
  • 

  And this then is the { Algebraic Identity on Complex Multiplication } we shall be using in Part IV / V / VI / VII of this paper .

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A Quick Review of a Special Type of Recipricals :

• Let us now take a quick look at a special type of Recipricals .

  We then bring in :

  • 

    

    

  And take note here that :

  • 

  • 

    and

  • 

  Consequently , we can now re-write the above equation as follows :

  • 

• And of course :
  • 

  We then have :

  • 

  consequently yielding this final equation :

  • 

• And the above equation is then the { Special Type of Reciprical Relation }
  • we shall be using in the next 2 Sections and beyond .

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What's Next :

• And we are now ready to solve for the 2 Equilateral Triangles , next .

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

go to the last Section : Section XXI - Solving the 3rd-Order Polynomial Equation

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07