On a Quick Expansion Method for Pythagorus Triples

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

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Section III - Outline and Purpose for Part III

Summary for the Section :

• In this Section , we shall outline what we intend to accomplish in this Part III of the paper .

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Recalling the Discovery :

• As said , we did happen to stumble upon the { Quick Expansion Procedure for Pythagorus Triples } ,
  • while doing the last paper :
    • On Breaking-down any Triangle into a Pair of Mirror-Imaged Equilateral Triangles

  As such , we must first do a brief recount of the relevant Sections of that last paper .

• In essence , we shall set up a triangle { Triangle A-B-C } on the Complex Plane ,
  • using 2 { Pythagorus-Triple Triangles } resized to the same height ,
    • and put into this configuration as shown below ;

    so that :

    • [ Point A ] will always lie on the { REAL-Axis } and positioned as such so that :
      • the centroid of { Triangle A-B-C } will always remain at [ Point O ] the Point-Of-Origin .

  

  We then assigned complex values to [ A ] / [ B ] / [ C ] according to their positions on the Complex Plane ,

  • and proceeded to solve for the 2 Mirror-Imaged Equilateral Triangles .

• In our solution procedure , we did assume that :
  • [ A ] / [ B ] / [ C ] are the solution values to a { 3rd-Order Polynomial Equation } of the format :
    • 

    so that these 2 equations below are assumed equivalent :

    • 

      and

    • 

  We then calculated the complex values of [ K2 ] / [ K1 ] / [ K0 ] based on the values of [ A ] / [ B ] / [ C ] :

  • via these relations below :
    • 

    • 

    • 

• Further down the solution procedure , we did set up the complex values [ W1 ] and [ W2 ] as follows :
  • 

  • 

    where :

    • 

    • 

  We then concentrated on the { value under the square-root sign } ,

  • and reverted to expressing this using complex values [ K1 ] and [ K0 ] :
    • 

      yielding :

    • 

  And in our calculation process for this { value under the square-root sign } ,

  • we did set up the complex value [ V-sub-IQ ] being the { Value-In-Question } , defined as :
    • 

  We then happen to make the observation that :

  • the [ Real Part ] and the [ Imaginary Part ] of the { Value-In-Question } do form a Pythagorus Triple .

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What's Next :

• Let us now set up the new { Triangle A-B-C } ,
  • enroute to our discovery of the Quick Pythagorus Triple Expansion Phenomenon in Section IX ;

    next .

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

go to the last Section : Section IV - The Usual Procedure for Expanding Pythagorus Triples

return to the Pythagorus Triples Homepage

Original dated 2021-12-27