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 LII - Special Comments on { Mirror-Imaged Regular Pentagons }

Summary for the Section :

• In this Section , we shall take a quick look at :
  • the usage of { Mirror-Imaged Regular Pentagons } in the solution to
    • certain restricted types of { 5th-Order Polynomial Equations } .

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Bringing in an Algebraic Identify :

• Let us first set up an Algebraic Identity for our use below .

  We then bring in these 3 equations involving the general complex variables [ T ] and [ U ] :

  • 

  • 

  • 

  Adding the 3 equations together then yields us this Algebraic Identity :

  • 

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A First Look at 5th-Order Polynomial Equations :

• Let us bring in a { 5th-Order Polynomial Equation } of the general format :
  • 

  Let us assume for the moment that :

  • the complex-values [ A ] / [ B ] / [ C ] / [ D ] / [ E ] are the solution values
    • to the above { 5th-Order Polynomial Equation } ;

  so that :

  • this equation below :
    • 

    and

  • the { 5th-Order Polynomial Equation } above :
    • 

    may now be considered equilvalent equations .

• We then have the relations :
  • 

  • 

  • 
    

  • 

  • 

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Putting on 3 Restrictions :

• Let us now list out the 3 restrictions we shall be imposing on the general { 5th-Order Polynomial Equation } :
  • FIRST RESTRICTION :
    • 

  • SECOND RESTRICTION :
    • 

  • THIRD RESTRICTION :
    • 

• Let us now look at the effects of these 3 Restrictions .

  FIRST RESTRICTION :

  • 

    Upon application of the this Restriction , the original { 5th-Order Polynomial Equation } :

    • 

    would become :

    • 

  SECOND RESTRICTION :

  • 

    Upon application of the this Restriction , the revised { 5th-Order Polynomial Equation } after the application of the FISRT RESTRICTION :

    • 

    would become :

    • 

    And re-arranging terms , we have this relation below :

    • 

  THIRD RESTRICTION :

  • 

    Further expansion on this then yields us this relation :

    • 

    Consequently :

    • 

  • Let us now introduce the fixed complex-value [ M ] :
    • 

    Consequently :

    • 

    Let us bring back from above this relation on [ K1 ] :

    • 

    Substituting therein the result immediately preceding then yields us this relation :

    • 

    Consequently :

    • 

    Let us bring back this equation from the SECOND RESTRICTION :

    • 

    Substituting therein these 2 results :

    • 

    • 

    then yields us this equation :

    • 

• Let us now introduce the fixed complex-value [ N ] :
  • 

  Consequently :

  • 

  Thus , we can now re-write the last { 5th-Order Polynomial Equation } above in this format :

  • 

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A Quick Comparison with the Algebraic Identity :

• Let us bring-in the above equation and the Algebraic Identity above for a quick comparison :
  • 

  • 

  Let us also bring in this { 3rd-Order Polynomial Equation } and the associated Algebraic Identity from Part IV as a reference :

  • 

  • 

  Striking similarity here, of course .

• Consequently , we can apply the same logical analysis here as in Part IV ,
  • to come up with these two (2) Conditions for solving the restricted { 5th-Order Polynomial Equation } here :

    i.e. :

    • Condition ONE :
      • 

    • Condition TWO :
      • 

  As a result , we can now proceed to solve for the pair of { mirror-imaged Regular Pentagons } ,

  • following the same procedure as we have outlined in Part IV of this paper .

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{ FENG-Line } for this restricted-type of { 5th-Order Polynomial Equation } :

• Let us bring back the FIRST RESTRICTION :
  • 

  Since :

  • 

    it then follows that :

    • 

  And what this means is that :

  • the Point-of-Origin [ Point O ] is the center-of-mass for a { 5-Equal-Mass System } of :
    • [ Mass-A ] / [ Mass-B ] / [ Mass-C ] / [ Mass-D ] / [ Mass-E ] ,

    regardless of whether :

    • [ Point O ] is serving as the intersection of the [ REAL-Axis ] and the [ IMAGINARY-Axis ] in a { complex plane } environment ,

      or

    • [ Point O ] is serving as the intersection of the [ X-Axis ] and the [ Y-Axis ] in a { 2-Dimensional Plane } environment .

  And to that extent ,

  • our analysis on { Invariancy upon Axis-Rotation } in Part V ,

  • our analysis on { Invariancy upon Resizing } in Part IV ,

    are also applicable here for the pair of { mirror-imaged Regular Pentagons } .

• Let us reformat this equation from above :
  • 

  into this new format :

  • 

  This then tell us that :

  • the { FENG-Line } and the { FENG-Circle } may also be applicable to :
    • the 2 { mirror-imaged Regular Pentagons } arising from this special type of restricted { 5th-Order Polynomial Equation } :
      • 

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What's next :

• Concluding Remarks , next .

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go to the next Section : Section LII - Concluding Remarks

go to the last Section : Section L - Special Comments on the { Equal-Mass Three-Body Problem }

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07