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 XXXI - [ T1 ] / [ T2 ] / [ T3 ] - OLD vs. NEW

Summary for the Section :

• In this Section , we shall establish the relations on [ T1 ] / [ T2 ] / [ T3 ] :
  • under the [ OLD Reference System ] and the [ NEW Reference System ] .

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[ W1 ] - OLD vs. NEW :

• Let us first set up these 2 equations on [ W1 ] for our comparison purposes here in Part V :
  • Under the { OLD Reference System } :
    • 

  • Under the { NEW Reference System } :
    • 

  We then bring in this result from the last Section - Section XXX :

  • 

  Consequently , we can now write :

  • 

• Let us bring back this { Algebraic Identity on Complex Multiplication } established previously in Section XXII :
  • 

  Consequently , we have :

  • 

  And on applying the { Algebraic Identity on Complex Multiplication } and the result immediately above to this equation from above :

  • 

  we then have :

  • 

  Let us now bring back this equation from above for a quick comparison :

  • 

  And we can see from the above 2 equations that :

  • 

    and

  • 

• We can then set up the fixed-value [ R-sub-T123 ] for our use below :
  • 

    for both the { OLD Reference System } and the { NEW Reference System } .

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

• We then set up these 6 equations on [ T1 ] / [ T2 ] / [ T3 ] for our comparison purposes :
  • Under the { OLD reference System } :
    • 

    • 

    • 

      where :

      • 

      • 

      • 

  • Under the { NEW Reference System } :
    • 

    • 

    • 

      where :

      • 

      • 

      • 

• Since we have already established above :
  • 

    we can now re-write these 3 equations from above :

    • 

    • 

    • 

    as :

    • 

    • 

    • 

• Let us now bring back these 3 equations from above :
  • 

  • 

  • 

  Substituting these therein to the 3 equations immediately preceding then yields us these 3 new equations :

  • 

  • 

  • 

• Consequently , we can now re-write these 3 original equations from above :
  • 

  • 

  • 

  as :

  • 

  • 

  • 

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A Graphic Look at [ T1 ] / [ T2 ] / [T3 ] :

• Based on the result immediately above , we can now summarize :
  • For the { OLD Reference System } :
    • 

    • 

    • 

  • For the { NEW Reference System } :
    • 

    • 

    • 

  

• And we see here that { Triangle T1-T2-T3 } is the same triangle for both under the { OLD Reference System } and the { NEW Reference System } .

  Essentially ,

  • when we rotate the { OLD X-Y Axis } clockwise thru the angle DELTA to arrive at the { NEW X-Y Axis } :
    • the end-result is simply that :
      • 
      • 
      • 

    so that :

    • the position vectors [ Vector O-T1 ] / [ Vector O-T2 ] / [ Vector O-T3 ] will remain pointing in the same directions as before .

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Preparing for the next Section :

• Let us bring back the last 3 equations immediately above :
  • 

  • 

  • 

  And based on this { Algebraic Identify on Complex Multiplication } established back in Section XXII :

  • 

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

    • 

    • 

    • 

  Consequently , we have :

  • 

  • 

  • 

  arising from :

  • 

  • 

  • 

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What's Next :

• And with this in hand , we are ready move on to solving for [ U1 ] / [ U2 ] / [ U3 ] , next .

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go to the next Section : Section XXXII - [ U1 ] / [ U2 ] / [ U3 ] - OLD vs. NEW

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

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07