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

Summary for the Section :

• In this Section , we shall establish the relations on [ U1 ] / [ U2 ] / [ U3 ] :
  • under the { OLD Reference System } and the { NEW Reference System } .

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Recalling the solution procedure :

• Let us recall from Section XXIII that :
  • Once we have established the values for [ T1 ] / [ T2 ] / [ T3 ] , we did obtain the corresponding values for the [ U ]'s via :
    • 

    We then set up :

    • [ U1 ] corresponding to [ T1 ] :
      • 

    • [ U2 ] corresponding to [ T2 ] :
      • 

    • [ U3 ] corresponding to [ T3 ] :
      • 

• And based on the above , let us now set up these 6 equations defining the [ U ]'s for our comparison purposes here in Part V :
  • Under the { OLD Reference System } :
    • 

    • 

    • 

  • Under the { NEW Reference System } :
    • 

    • 

    • 

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Recalling [ M ] and [ T1 ] / [ T2 ] / [ T3 ] - OLD vs. NEW :

• For [ M ] , let us bring in again this relation previously established in Section XXIX :
  • 

  And for the [ T ]'s , let recall that we did establish this 3 relations below at the end of the last Section - Section XXXI :

  • 

  • 

  • 

• We then bring back these 3 equations from above :
  • 

  • 

  • 

  and apply thereto the 4 relations immediately preceding to yield :

  • 

  • 

  • 

• Since we have defined above :
  • 

  • 

  • 

  we can now re-write the 3 equations immediately preceding above as :

  • 

  • 

  • 

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Setting up U1 / U2 / U3 - OLD vs. NEW :

• Let us recall that we did establish in the last Section - Section XXX that :
  • 

  And based on :

  • 

  • 

    we did also establish that :

    • 

• We then set up the fixed value { R-sub-T-123 } :
  • 

  so that :

  • we were able to set up the 6 [ T's ] as follows :
    • for the { OLD Refernce System } :
      • 

      • 

      • 

    • for the { NEW Reference System } :
      • 

      • 

      • 

• Let us recall from above that :
  • 

  And based on a quick comparison of :

  • 

  • 

    we can now conclude that :

    • 

  Recalling that :

  • 

    we can now set up the fixed value { R-sub-U-123 } :

    • 

    so that :

    • we can now set up the 6 [ U's ] as follows :
      • under the { OLD Reference System } :
        • 

        • 

        • 

      • under the { NEW Reference System } :
        • 

        • 

        • 

• Let us now bring in these results already established above :
  • 

  • 

  • 

  Substituting therein the values of the OLD [ U ]'s we have just established above ;

  • and also taking note of this { Algebraic Identity on Complex Multiplication } established in Section XXII :
    • 

  we then have these final relations for the NEW [ U ]'s below :

  • 

  • 

  • 

  This then tells us that :

  • Under the { NEW Reference System } :
    • { Triangle U1-U2-U3 } remained in a Equilateral-Triangle formation ;

      and

    • [ Vector O-U1 ] remained pointing in the same direction as before ,

    • [ Vector O-U2 ] remained pointing in the same direction as before ,

    • [ Vector O-U3 ] remained pointing in the same direction as before ;

    even though the { X-Y Axis } have been rotated clockwise thru the angle [ Delta ] .

  X

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A Graphic Look at [ U1 ] / [ U2 ] / [ U3 ] :

• Based on the results immediately preceding , we then have this set of 3 diagrams below :
  • showing the [ U ]'s before and after the [ X-Y Axis Rotation ] .

  

• And we see here that { Triangle U1-U2-U3 } is the same triangle for under the { OLD Reference System } and the { NEW Reference System } .

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What's Next :

• Let us summarize our findings here in Part V , next .

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go to the next Section : Section XXXIII - Summarizing on Part V

go to the last Section : Section XXXI - [ T1 ] / [ T2 ] / [ T3 ] - OLD vs. NEW

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07