On a Quick Expansion Method for Pythagorus Triples

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

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Section X - Outline and Purpose for Part IV

Summary for the Section :

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

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Outline for Part IV :

• Given our first success with the { Quick Expansion Method for Pythagorus Triples } in the last Section - Section IX ,
  • it is necessary for us to find out if the procedure works , possibly all the way thru to infinity .

  In this Part IV of the paper , we shall test the procedure via bringing in :

  • the first 5 Pythagorus Triples , namely :
    • the { 3-4-5 Pythagorus Triple } ,

    • the { 5-12-13 Pythagorus Triple } ,

    • the { 8-15-17 Pythagorus Triple } ,

    • the { 20-21-29 Pythagorus Triple } , and

    • the { 12-35-37 Pythagorus Triple } ;

  so that :

  • 10 pairs of base { Right-Angle Triangles } may be formed therefrom ,
    • for the purpose of constructing the new { Triangle A-B-C }'s ;

    i.e. :

    • the { 3-4-5 / 5-12-13 Pair } ,

    • the { 3-4-5 / 8-15-17 Pair } ,

    • the { 3-4-5 / 20-21-29 Pair } ,

    • the { 3-4-5 / 12-35-37 Pair } ,

    • the { 5-12-13 / 8-15-17 Pair } ,

    • the { 5-12-13 / 20-21-29 Pair } ,

    • the { 5-12-13 / 12-35-37 Pair } ,

    • the { 8-15-17 / 20-21-29 Pair } ,

    • the { 8-15-17 / 12-35-37 Pair } , and

    • the { 20-12-29 / 12-35-37 Pair } .

• And for each of the above 10 Pairs of base Triangles ,
  • we will always present the full results from all 4 combinations of orientations possible .

  Let us explain a bit more on this via this example for the { 3-4-5 / 5-12-13 Pair } ,

  • as per this set of 4 diagrams below .

    
    

    We always start with :

    • the orientation as per the diagram on the Top-Left ,
      • matching the [ 4 ] in the { 3-4-5 Triangle } and the [ 12 ] in the { 5-12-13 Triangle }
        • so that the height of both triangles will be the same .

    We then move over to :

    • the orientation as per the diagram on the Top-Right ,
      • matching the [ 4 ] in the { 3-4-5 Triangle } and the [ 5 ] in the { 5-12-13 Triangle }
        • so that the height of both triangles will be the same .

    We then move down to :

    • the orientation as per the diagram on the Bottom-Right ,
      • matching the [ 3 ] in the { 3-4-5 Triangle } and the [ 5 ] in the { 5-12-13 Triangle }
        • so that the height of both triangles will be the same .

    And finally , we move left-wards to :

    • the orientation as per the diagram on the Bottom-Left ,
      • matching the [ 3 ] in the { 3-4-5 Triangle } and the [ 12 ] in the { 5-12-13 Triangle }
        • so that the height of both triangles will be the same .

• The presentation order for the { height matching procedure for the 2 base Triangles }is then :
  • [ even-on-even ] / [ even-on-odd ] / [ odd-on-odd ] / [ odd-on-even ] .

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What's Next :

• In the next 10 Sections , we shall present the results arising from the above-said 10 Pairs of base Triangles .

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go to the next Section : Section XI - Results for the [ 3-4-5 / 5-12-13 ] Pythagorus-Triple Pair

go to the last Section : Section IX - Setting up the { Value-In-Question } - [ V-sub-IQ ]

return to the Pythagorus Triples Homepage

Original dated 2021-12-27