On a Quick Expansion Method for Pythagorus Triples

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

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Section XXXI - Developing the One-On-One Expansion Formula ( I )

Summary for the Section :

• In this Section , we shall express the { Value-In-Question } :
  • 

    in terms of [ W ] , [ D ] , and [ H ] .

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Setting up [ K1 ] and [ K0 ] :

• Let us being in the Complex Values [ A ] / [ B ] / [ C ] from the last Section :
  • 

  • 

  • 

  Consequently , we have :

  • First :
    • 

  • Secondly :
    • 

    • 

    • 

      yielding :

      • 

  • Third-and-Finally :
    • 

• Let us recall that :
  • In order to solve for the 2 mirror-imaged Equilateral Triangles associated with the given { Triangle A-B-C } ,
    • we shall assume that the complex values [ A ] / [ B ] / [ C ] are the solution values
      • for a { 3rd-Order Polynomial Equation } of the format :
        • 

      so that :

      • these 2 equations below shall be considered equivalent throughout this Section here .
        • 

        • 

  As such :

  • upon expansion of the 2nd equation above , we have :
    • 

    • 

    • 

• Let us recall that we have established above :
  • 

  • 

  • 

  Therefore :

  • 

  • 

  • 

  Consequently we can now re-write the original { 3rd-Order Polynomial Equation } ,

  • i.e. this equation :
    • 

    in this revised format :

    • 

  And based on this , we can now proceed to express the complex value for the { Value-In-Question } , i.e. :

  • 

    in terms of [ W ] / [ D ] / [ H ] ,

    • using the procedure we have previously established in Part III of this paper .

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Doing the calculations for [ K1 ] :

• We then bring-in [ K1 ] from above :
  • 

• Then , for the { square of [ K1 ] } , we have :
  • 
    

  And on consolidating terns , we have :

  • 
    

• Then , for the { cube of [ K1 ] } , we have :
  • 
    
    
    
    
    
    
    
    
    

  And on consolidating terms , we have :

  • 
    
    
    
    

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Doing the calculations for [ K0 ] :

• We then bring-in [ K0 ] from above :
  • 

• Then , for the { square of [ K0 ] } , we have :
  • 
    
    

  And on consolidating terms , we have :

  • 
    
    

  Consequently , we have :

  • 
    
    
    

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Calculating the { Value-In-Question } :

• Let us recall { V-sub-IQ } from above :
  • 

  We then bring in from above :

  • 
    
    
    

  and

  • 
    
    
    
    

  We then have :

  • 
    
    
    
    

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Setting up [ P ] and [ Q ] :

• Let us again set up [ V-sub-IQ ] in this special format :
  • 

    where :

    • [ P ] is the real-part of [ V-sub-IQ ] ,

      and

    • [ Q ] is the imaginary-part of [ V-sub-IQ ] .

  We then have , from immediately above :

  • 
    
    

  and

  • 
    
    

• And on re-arranging terms , we have this final format for [ P ] and [ Q ] :
  • 
    

  and

  • 
    

    for our further analysis in the next Section .

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What's Next :

• Let us now take a closer look at [ P ] and [ Q ] , next .

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go to the next Section : Section XXXII - Developing the One-On-One Expansion Formula ( II )

go to the last Section : Section XXX - Setting up the new { Triangle A-B-C } for the One-On-One Expansion

return to the Pythagorus Triples Homepage

Original dated 2021-12-27