On a Quick Expansion Method for Pythagorus Triples

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

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Section VII - Creating the 3rd-Order Polynomial Equation from { Triangle A-B-C }

Summary for the Section :

• In this Section , we shall create a { 3rd-Order Polynomial Equation }
  • based on { Triangle A-B-C } established in the last Section - Section VI .

  This is a necessary step towards the solving for the 2 { Mirror-Imaged Equilateral Triangles } .

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Converting to the Complex Plane :

• Let us now convert the { X-Y Plane } into a { Complex Plane } via :
  • converting the { X-Axis } into the { REAL-Axis } ,

    and

  • converting the { Y-Axis } into the { IMAGINARY-Axis } ;

    but :

    • without any changes to the { unit circle } or the distance-calibrations ;
      • as per this set of 3 diagrams below .

  

  

• As such , we can now assign complex values to [ Point A ] , [ Point B ] and [ Point C ] ,
  • based on their respective positions on the { Complex Plane } .

  In particular :

  • [ Point A ] will now be assigned the complex value [ A ] ,

  • [ Point B ] will now be assigned the complex value [ B ] ,

  • [ Point C ] will now be assigned the complex value [ C ] .

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Bringing in a 3rd-Order Polynomial Equation :

• For our analysis purposes here in the 3 Sections ,
  • we shall now assume that [ A ] , [ B ] and ] C ] are the solution values
    • for a particular 3rd-Order Polynomial Equation of the format :
      • 

  And that is to say that :

  • These 2 equations below are to be considered equivalent equations :
    • 

    • 

      throughout our analysis here in the 3 Sections .

  And on expansion of the second equation above and comparing it with the first equation ,

  • we then have :
    • 

    • 

    • 

• Since [ Point O ] is the centroid of { Triangle A-B-C } ,
  • we always have :
    • [ Vector O-A ] + [ Vector O-B ] + [ Vector O-C ] = [ zero ]

  This then lead us to conclude that :

  • the complex values [ A ] , [ B ] and [ C ] will also add up to [ zero ] , i.e. :
    • 

• Consequently , the expression :
  • 

    will also yield us a [ zero ] value , i.e. :

    • 

• Thus , we can re-write our original 3rd-Order Polynomial Equation :
  • 

  as :

  • 

  And on re-arranging terms , we have :

  • 

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What's Next :

• And we shall take a quick look at the Solution Procedure for this equation , next .

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go to the next Section : Section VIII - Solving the 3rd-Order Polynomial Equation

go to the last Section : Section VI - Setting up the Random Triangle and its Reference Frame

return to the Pythagorus Triples Homepage

Original dated 2021-12-27