On a Quick Expansion Method for Pythagorus Triples

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

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Summary and Key Findings :

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Summary :

• In this paper , we shall develop 2 quick Expansion Formula for Pythagorus Triple :
  • the Double-Up Expansion Formula :
    • which involves only one ( 1 ) base { Pythagorus-Triple Triangle } ; and

  • the One-On-One Expansion Formula :
    • which involves two ( 2 ) base { Pythagorus-Triple Triangles } ,
      • such as the { 3-4-5 Triangle } vs. the { 5-12-13 Triangle } .

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Key Finding One :

• We first developed the { Double-Up Expansion Formula } applicable to any Pythagorus Triple .

  For any Pythatgorus Triple { W / H / L } obeying :

  • 

  we can always set up another Pythagorus Triple , with :

  • 

  • 

  • 

  so that :

  • 

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Key Finding Two :

• We also developed the { One-On-One Expansion Formula } for combining 2 { Pythagorus Triples } :

  We first bring in 2 { Pythagorus-Triple Triangles } resized to the same height ,

  • and re-position the 2 triangles on the [ complex plane ] as per this set of 3 diagrams below :

  

  so that :

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

  We then set up the key values [ W ] / [ D ] / [ H ] via :

  • 

  • 

  • 

  so that :

  • the assigned complex values of [ A ] / [ B ] / [ C ] are then :
    • 

    • 

    • 

  And based on this , we then developed the { One-On-One Expansion Formula } for the Pythagorus Triples ,

  • and the resultant { [ X ] / [ Y ] / [ Z ] Pythagorus Triple } is then :
    • 

    • 

    • 

• This looks rather complicated and we were able to revised this into this format below :
  • 

  • 

  • 

    at the end of Part VI of the paper .

Table of Content

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Original dated 2017-12-27