On Breaking-down any Triangle into
a pair of Mirror-Imaged Equilateral Triangles

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

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Section XXXIX - Summarizing on Part VI

Summary for the Section :

• In this Section , we shall summarize our findings for Part VI .

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Recalling the Start-Off Position :

• Let us recall that started off this Part VI with the resizing of the unit circle :

  

  We then set up the scalar quantity [ PSI ] :

  • 

  The relations for the complex values [ A ] / [ B ] / [ C ] :

  • under the { OLD System } vs. under the { NEW System }

    are then given by :

    • 

    • 

    • 

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[ T ] / [ U ] / [ X ] - OLD vs. NEW :

• We then established :
  • for [ T1 ] / [ T2 ] / [ T3 ] :
    • 

    • 

    • 

  • for [ U1 ] / [ U2 ] / [ U3 ] :
    • 

    • 

    • 

  • for [ X1 ] / [ X2 ] / [ X3 ] :
    • 

    • 

    • 

  This then tell us that :

  • when we do the resizing by the scalar factor [ Psi ] :
    • all 4 triangles are resized in unison by the same factor ,

      i.e. , the 4 triangles :

      • { Triangle A-B-C } ,

      • { Triangle X1-X2-X3 } ,

      • { Triangle T1-T2-T3 } ,

      • { Triangle U1-U2-U3 } .

  

  

  And that is to say :

  • Nothing changes upon resizing .

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What's next :

• Let us now move on to Part VII of the paper ,
  • to look at an Invariant Line associated with each-and-every triangle other than the Equilateral Triangle ;

    next .

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go to the next Section : Section XXXX - Outline and Purpose for Part VII

go to the last Section : Section XXXVI - [ T ] / [ U ] / [ X ] - OLD System vs. NEW System

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07