On a Quick Expansion Method for Pythagorus Triples

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

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

Summary for the Section :

• In this Section , we shall set up the new { Triangle A-B-C } for the One-On-One Expansion Process ,
  • and define the complex values [ W ] , [ D ] and [ H ] associated with the said { Triangle A-B-C } ,
    • for our use in later Sections to come in this Part VI of the paper .

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Setting up the new { Triangle A-B-C } :

• Let us bring in any 2 { Pythagorus-Triple Triangles } :
  • as per the 2 diagrams on-the-left below .

  We then resize the 2 said Triangles so that their heights will now be the same , i.e. :

  • 

    as per the 2 diagrams on-the-right below .

  

  

• We then set up the new { Triangle A-B-C } ,
  • using the 2 resized { Pythagorus-Triple Triangles } above and the procedure as detailed in Part III of the paper ,
    • as per the diagram on-the-left above .

• And once we have set up the new { Triangle A-B-C } ,
  • we can then set up the 3 key complex values [ W ] , [ D ] and [ H ] :
    • 

    • 

    • 

      as per the diagram in-the-middle above .

    consequently yielding these 2 relations below :

    • 

    • 

  As such , the complex values for [ A ] / [ B ] / [ C ] can now be re-written as :

  • 

  • 

  • 

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A Quick Example for Illustration Purposes :

• Let us now do a quick example for setting up the new { Triangle A-B-C } for the sake of clarity .

  We then bring in the { 20-21-29 Triangle } and the { 12-35-37 Triangle }

  • as per the 2 diagrams on-the-left below .

  And the 2 said Triangles are then resized so that :

  • the height of the 2 Triangles are now the same , at [ 105 ] ;
    • as per the 2 diagrams on-the-right below .

  

  

• We then set up the new { Triangle A-B-C } via the procedure as detailed in Part III of this paper ,
  • as per the diagram on-the-left above .

  Consequently , the complex values [ A ] / [ B ] / [ C ] here are then :

  • 

  • 

  • 

• We then have these complex values below for [ W ] , [ D ] and [ H ] arising from { Triangle A-B-C } :
  • 

  • 

  • 

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What's Next :

• And with this in-hand , we are now ready for the development of the One-On-One Expansion Formula , next .

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go to the next Section : Section XXXI - Developing the One-On-One Expansion Formula ( I )

go to the last Section : Section XXIX - Outline and Purpose for Part VI

return to the Pythagorus Triples Homepage

Original dated 2021-12-27