On a Quick Expansion Method for Pythagorus Triples

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

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Section XXIII - Setting up the new { Triangle A-B-C } for the Doubling-Up Process

Summary for the Section :

• In this Section , we shall set up the new { Triangle A-B-C } for the Doubling-Up Process .

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Setting up the new { Triangle A-B-C } for the Doubling-Up Process :

• Let us first bring-in the { 3-4-5 Pythagorus-Triple Triangle } ,
  • and identify the angle { Angle THETA } as being the { arctangent of [ 3-over-4 ] } ,
    • as per the diagram on-the-left below .

  Consequently , we can set up the new { Triangle A-B-C } in accordance to procedure as described in Part III of this paper ,

  • but now using 2 { 3-4-5 Pythagorus-Triple Triangles } instead ;
    • as per the diagram in-the-middle below .

  

  But we take note at this point that :

  • { Triangle B-O-C } is a { 15-20-25 Pythagorus-Triple Triangle } ,
    • with [ Angle B-O-C ] being a right-angle of 90 degrees ;
      • as per the diagram on-the-right above .

• This then allowed us to do things more simply via :
  • rotating the new { Triangle A-B-C } counter-clockwise thru [ Angle THETA ] ,
    • to arrive at re-oriented { Triangle A-B-C } as per the diagram in-the-middle below .

  

  We then further simplified things via re-sizing { Triangle A-B-C }

  • so that the final values for [ A ] / [ B ] / [ C ] are then :
    • [ A ] = + 4 + 3i

    • [ B ] = - 4

    • [ C ] = - 3i

      as per the diagram on-the-right above .

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What's Next :

• And we shall do the calculations for the Value-In-Question [ V-sub-IQ ] based on the above , next .

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go to the next Section : Section XXIV - First Try at the Doubling-Up Process

go to the last Section : Section XXII - Outline and Purpose for Part V

return to the Pythagorus Triples Homepage

Original dated 2021-12-27