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 XLVIII - The { FENG-Line } and { FENG-Circle } in action

Summary for the Section :

• In this Section , we shall explain why :
  • the { FENG-Line } and the { FENG-Circle } are important structural features of the Triangle .

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Bringing in the base { Triangle A-B-C } before making changes :

• Let us suppose , for the moment , that :
  • we were successful in breaking-down a random triangle { Triangle A-B-C }
    • into its 2 component equilateral triangles { Triangle T1-T2-T3 } and { Triangle U1-U2-U3 } respectively ,

    as per this set of 6 diagrams below :

  

  

  Let us also draw in the { FENG-Line } and the { FENG-Circle ) as per this set of 3 diagrams below .

  

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Rotating { Triangle T1-T2-T3 } and { Triangle U1-U2-U3 } in opposite directions :

• Let us now do the following :
  • rotate { Triangle T1-T2-T3 } in the counter-clockwise direction using [ Point O ] as the pivotal-point ,
    • thru an angle { epsilon } ;

  • rotate { Triangle U1-U2-U3 } in the clockwise direction using [ Point O ] as the pivotal-point ,
    • thru the same angle { epsilon } ;

  • create a { NEW Triangle A-B-C } via the relations :
    • [ NEW Vector O-A ] = [ NEW Vector O-T1 ] + [ NEW Vector O-U1 ]

    • [ NEW Vector O-B ] = [ NEW Vector O-T2 ] + [ NEW Vector O-U2 ]

    • [ NEW Vector O-C ] = [ NEW Vector O-T3 ] + [ NEW Vector O-U3 ]

      as per the set of 6 diagrams below .

  

  

• Interesting enough :
  • the { FENG-Line } for the { NEW Triangle A-B-C } will remain in the same position as before .

  This is because :

  • the angle-bisector of { angle T1-O-U1 } / { angle T2-O-U2 } / { angle T3-O-U3 }
    • will be the same { FENG-Line } for both the { OLD Triangle A-B-C } and the { NEW Triangle A-B-C } .

      ( we have rotated { Triangle T1-T2-T3 } and { Triangle U1-U2-U3 } in opposite direction thru the same angle . )

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Resizing { Triangle T1-T2-T3 } and { Triangle U1-U2-U3 } :

• Let us now resize { Triangle T1-T2-T3 } and { Triangle U1-U2-U3 } but without changing their orientations ,
  • while making sure that this relation below would remain in tact :
    • 

  so that :

  • the { NEW Triangle A-B-C } will then be created via these relations :
    • [ NEW Vector O-A ] = [ NEW Vector O-T1 ] + [ NEW Vector O-U1 ]

    • [ NEW Vector O-B ] = [ NEW Vector O-T2 ] + [ NEW Vector O-U2 ]

    • [ NEW Vector O-C ] = [ NEW Vector O-T3 ] + [ NEW Vector O-U3 ]

      as per the set of 6 diagrams below .

  

  

• And we take special note here that :
  • the { FENG-Line } will remain in tact because :
    • the orientations of { Triangle T1-T2-T3 } and { Triangle U1-U2-U3 } have remained unchanged ;

  • the { FENG-Circle } will also remain in tact because :
    • this relation below remains valid for newly created { Triangle T1-T2-T3 } and { Triangle U1-U2-U3 } :
      • 

      so that :

      • the values of [ M ] / [ PHI-1 ] / [ PHI-2 ] would have remained constant / unchanged throughout in this relation below :
        • 

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Summarizing on the { FENG-Line } :

• Let us now bring in the 3 { Triangles A-B-C }'s established above for a quick comparison :
  • with the [ base Triangle A-B-C ] being in-the-middle in this set of 3 diagrams below .

  

  In essence :

  • the { FENG-Line } is a controlling feature on the orientation of { Triangle 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 :

• In the next 2 Sections , we shall demonstrate that :
  • the { FENG-Line } is in fact a { Pricipal Axes } for { 3-equal-mass System } .

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go to the next Section : Section XLIX - Principal Axes for a { 3-Equal-Mass System }

go to the last Section : Section XLVII - Explaining the { FENG-Line } and the { FENG-Circle }

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07