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 XXXVIII - [ T ] / [ U ] / [ X ] - OLD System vs. NEW System

Summary for the Section :

• In this Section , we shall establish the relations for [ T ] / [ U ] / [ X ] :
  • under the { OLD System } vs. under the { NEW System } .

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Recalling the Solution Procedure for [ T ] :

• Let us recall that the next step in our solution procedure for the 2 Equilateral Triangles was to solve for [ T ] :
  • via this equation below :
    • 

  We then identified [ T1 ] / [ T2 ] / [ T3 ] as the cube-root of [ W1 ] ,

  • so that :
    • 

  Since we have already established in the last Section - Section XXXVII :

  • 

  it then follows :

  • 

  • 

  • 

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[ U1 ] / [ U2 ] / [ U3 ] - OLD vs. NEW :

• Let us recall that the corresponding [ U ]'s are given by :
  • 

  Since we have established in the last Section - Section XXXVII :

  • 

    and also established above :

    • 

    • 

    • 

  it then follows that :

  • 

  • 

  • 

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[ X1 ] / [ X2 ] / [ X3 ] - OLD vs. NEW :

• Let us recall that we then established the corresponding [ X ]'s via :
  • 

  And based on these 6 relations established above :

  • 

  • 

  • 

    and

  • 

  • 

  • 

  we can now write :

  • 

  • 

  • 

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What's next :

• Let us summarize our findings for Part VI , next .

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go to the next Section : Section XXXIX - Summarizing on Part VI

go to the last Section : Section XXXVII - [ W1 ] and [ W2 ] - OLD System vs. NEW System

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07