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 XXXXVII - Explaining the { FENG-Line } and the { FENG-Circle }

Summary for the Section :

• In this Section , we shall explain :
  • why the { FENG-Line } is unique to every triangle other than the { equilateral triangle } .

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The { PHI-1 } and { PHI-2 } Vectors :

• Let us recall that we did set up , in Section XLIV , [ PHI-1 ] and [ PHI-2 ] as the 2 different-and-distinct square-roots of [ M ] :
  • i.e. :
    • 

    • 

  As such ,

  • the { position vector for [ PHI-1 ] } and the { position vector for [ PHI-2 ] }
    • will point in the exact-opposite directions on the Complex Plane .

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Invariance of [ PHI-1 ] and [ PHI-2 ] upon { X-Y Axis Rotation } :

• Let us also recall that in Part V of this paper , we did set up 2 sets of { X-Y Axis } , namely :
  • the { OLD Set of X-Y Axis } and { NEW Set of X-Y Axis } differing by the rotation angle { Delta } ,
    • for the purpose of investigating whether or not :
      • the 2 mirror-imaged triangles { Triangle T1-T2-T3 } and { Triangle U1-U2-U3 }
        • would move because of this type of { X-Y Axis Rotation } .

    as per this set of 3 diagrams below .

  

  Consequently , we did establish under those circumstances :

  • 

  Since [ PHI-1 ] and [ PHI-2 ] are the square-root of [ M ] , we then have :

  • 

  • 

• Let us now set up the [ PHI ]'s in the usual complex number format :
  • Under the { OLD Reference System } :
    • 

    • 

  • Under the { NEW Reference System } :
    • 

    • 

  Then , based on these 2 relations established above :

  • 

  • 

  we can now write :

  • 

  • 

    based again on this relation previously established in Section XXII :

    • 

  This then tell us that :

  • the position vectors for [ PHI-1 ] and [ PHI-2 ] will always point in the same opposite directions ;
    • regardless of how we might rotate the { X-Y Axis } and correspondingly the { REAL-COMPLEX Axis } .

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Invariance of [ PHI-1 ] and [ PHI-2 ] upon Resizing :

• Let us recall that we did set up , in Part VI of this paper , 2 differently sized { Unit Circle }'s :
  • for the purpose of investigating whether or not :
    • the 2 mirror-imaged triangles { Triangle T1-T2-T3 } and { Triangle U1-U2-U3 }
      • would change in size because of this type of { changing the distance-calibrating base } ;

    as per this set of 3 diagrams below .

  

  And we also set up the scalar value [ PSI ] being :

  • 

  Consequently , we did establish under those circumstances :

  • 

  Since [ PHI-1 ] and [ PHI-2 ] are the square-roots of [ M ] , it then follows that :

  • 

  • 

• This then tells us that :
  • when we do distance-recalibrating via the scalar quantity [ PSI ] :
    • { Triangle A-B-C } / { Triangle T1-T2-T3 } / { Triangle U1-U2-U3 }

      and

    • [ PHI-1 ] and [ PHI-2 ]

      will all be re-calibrated accordingly but without any relative size-changes .

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Establishing the { FENG-Line } :

• Based on the above , we can therefore establish the { FENG-Line } :
  • as the line being in the [ PHI-1 ] / [ PHI-2 ] direction .

• Let us now recall we did set up [ PHI-1 ] and [ PHI-2 ] as the square-root of [ M ] :
  • 

  • 

  We can then set up [ PHI-1 ] and [ PHI-2 ] in the usual complex-number format :

  • 

  • 

  Consequently , we can draw-in a circle of radius [ R-sub-PHI ] with [ Point O ] as the center-of-the-circle ,

  • and we shall label this circle as the { FENG-Circle } .

  

• Very simply :
  • the { FENG-Line } is a controlling feature on the orientation of { Triange A-B-C } ,

    and

  • the { FENG-Circle } is a controlling feature on the size of { Triangle A-B-C } .

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What's next :

• We shall take a look at why the { FENG-Line } and the { FENG-Circle } are important structural features of the Triangle , next .

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go to the next Section : Section XLVIII - The { FENG-Line }and { FENG-Circle } in action

go to the last Section : Section XLVI - Solving for [ Tau ] and [ Mu ]

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07