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 XLIX - Principal Axes for a { 3-Equal-Mass System }

Summary for the Section :

• In this and the next Section , we shall determine whether or not ,
  • the { FENG-Line } is indeed a Principal Axes of Rotation for a { 3-Equal-Mass System } of [ Mass-A ] / [ Mass-B ] / Mass-C ] .

  This Section then sets up the preliminaries for the above-said determination .

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Setting up the X-Y Axis based on the { FENG-Line } :

• Let us suppose for the moment that :
  • we are successful in breaking-down the 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 3 diagrams below .

  

  

• We can then set up a { set of orthogonal X-Y axis } with :
  • the X-Axis being in the { FENG-Line } direction

    and

  • the Y-Axis being in the direction orthogonal to the X-Axis ;
    • as per the set of 3 diagrams above .

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Setting up the Position Vectors :

• Let us now set up for { Triangle T1-T2-T3 } :
  • the distance fixed-value [ R-sub-T ]

    and

  • the angles [ Theta-T1 ] / [ Theta-T2 ] / [ Theta-T3 ] ;
    • as per this set of 4 diagrams below .

  

• Consequently , we have :
  • For [ Point T1 ] , the X-Y co-ordinates are given by :
    • 
    • 

  • For [ Point T2 ] , the X-Y co-ordinates are given by :
    • 
    • 

  • For [ Point T3 ] , the X-Y co-ordinates are given by :
    • 
    • 

  And since { Triangle T1-T2-T3 } is an equilateral triangle ,

  • we can express the angles [ Theta-T2 ] and [ Theta-T3 ] in terms of [ Theta-T1 ] via :
    • 

    • 

• Let us now also set up for { Triangle U1-U2-U3 } :
  • the distance fixed-value [ R-sub-U ]

    and

  • the angles [ Theta-U1 ] / [ Theta-U2 ] / [ Theta-U3 ] ;
    • as per this set of 4 diagrams below .

  

• Consequently , we have :
  • For [ Point U1 ] , the X-Y co-ordinates are given by :
    • 
    • 

  • For [ Point U2 ] , the X-Y co-ordinates are given by :
    • 
    • 

  • For [ Point U3 ] , the X-Y co-ordinates are given by :
    • 
    • 

• We take special note at this point that :
  • the angles are measured as [ counter-clockwise rotations from the X-Axis ] .

  And as such ,

  • Angle [ Theta T1 ] / [ Theta T2 ] / [ Theta-T3 ] will have positive values ,

  whereas :

  • Angle [ Theta U1 ] / [ Theta U2 ] / [ Theta-U3 ] will have negative values .

• Since the X-Axis is in fact the { FENG-Line } , we have these special relations below :
  • 

  • 

  • 

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The Position Vectors for { Triangle A-B-C } :

• Let us now take a look at the X-Y co-ordinates for [ Point A ] / [ Point B ] / [ Point C ] based on :
  • [ Vector O-A ] = [ Vector O-T1 ] + [ Vector O-U1 ]

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

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

  We then have :

  • For [ Point A ] , the X-Y co-ordinates are given by :
    • 
    • 

  • For [ Point B ] , the X-Y co-ordinates are given by :
    • 
    • 

  • For [ Point C ] , the X-Y co-ordinates are given by :
    • 
    • 

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Determining the Principal Axes :

• In order to determine whether or not :
  • the X-Axis is indeed a Principal Axes for a { 3-Equal-Mass System } of [ Mass-A ] / [ Mass-B ] / [ Mass-C ] ;

  it is necessary for us :

  • to set up the value [ Lamda ] , being given by :
    • 

  If , in fact , the value of [ Lamda ] turns out to be [ zero ] , then :

  • the [ X-Direction ] and consequently the { FENG-Line } is a Principal Axes for a { 3-Equal-Mass System } .

• Let us now do the calculations .

  We then have :

  • 
    

  • 
    

  • 
    

  Consequently :

  • 

  • 

  • 

• Taking note here again that :
  • 

  • 

  • 

  we can then re-write the 3 equations immediately preceeding as follows :

  • 

  • 

  • 

• Let us now bring in this set of 2 relations from above :
  • 

  • 

  We can then re-express the 3 equations immediately above using { angle Theta-T1 } as the base angle :

  • 

  • 

  • 

  Consequently , we have this final expression for [ Lamda-XY ] :

  • 

    arising from :

    • 

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What's next :

• Let us now take a look as to whether or not :
  • 

    is true ; next .

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go to the next Section : Section L - Confirming the { FENG-Line } as a Principal Axes of Rotation

go to the last Section : Section XLVII - The { FENG-LIne } and { FENG-Circle } in action

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07