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 XXVII - Setting up the 2 different Reference Systems

Summary for the Section :

• In this Section , we shall set up 2 different Reference Systems , namely :
  • the { OLD Reference System } vs. the { NEW Reference System } ,
    • in preparation for our further analysis in this Part V of the paper .

Recalling the Random Triangle { Triangle A-B-C } :

• Let us first bring back the random triangle { Triangle A-B-C } we have created back in Section XIX ;
  • with the centroid of the triangle [ Point O ] being identified as the primary Point-Of-Referernce so that :
    • the position vectors [ Vector O-A ] , [ Vector O-B ] and [ Vector O-C ] will fully define the Triangle .

  
  

  Let us recall that we also did set up in that Section :

  • a { set of 2 orthogonal X-Y Axis } and a { unit-circle } so that :
    • the Position Vectors { Vector O-A } , { Vector O-B } and { Vector O-C } can be fully quantified ;
      • as per the set of 3 diagrams immediately above .

• And in this Part V of the paper , we shall refer to this existing referencing system as the { OLD Reference System } .

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Setting up the 2nd set of { orthogonal X-Y Axis } for our comparison purposes :

• Let us bring-in again the { OLD set of 2 orthogonal X-Y Axis } as per the diagram on-the-left below ,
  • and rotate the said { set of 2 orthogonal X-Y Axis } clockwise thru a random angle DELTA :
    • to arrive at a new { set of 2 orthogonal X-Y Axis } as per the diagram on-the-right below ;

    so that :

    • the newly created { set of 2 orthogonal X-Y Axis } shall be refered to as the { NEW Reference System }
      • throughout the remainder of this Part V of the paper .

  
  

• Let us also create correspondingly :
  • the { OLD Complex Plane } corresponding to the { OLD Reference System } ,
    • as per the diagram on-the-left above ;

    and

  • the { NEW Complex Plane } corresponding to the { NEW Reference System } ,
    • as per the diagram on-the-right above .

  And we take note here that the { Unit Circle } has remained unchanged ,

  • i.e. : same as before ,

  so that :

  • the distance [ R-sub-A ] / [ R-sub-B ] / [ R-sub-C ] are also the same as before .

• Let us now take a closer look at the assigned complex values of [ A ] / [ B ] / [ C ] :
  • under the { OLD Reference System } vs. the { NEW Reference System } .

  
  

  We then have , based on the 8 diagrams above :

  • For the { OLD Reference System } :
    • 

    • 

    • 

  • For the { NEW Reference System } :
    • 

    • 

    • 

  And the relations between the { OLD angles } and the { NEW angles } are then :

  • 

  • 

  • 

• And these then are the core 9 relations we shall working with throughout the remainder of this Part V of the paper .

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What's Next :

• And with this in hand , we are ready to do the analysis , next .

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go to the next Section : Section XXVIII - Complex Values [ A ] / [ B ] / [ C ] - OLD vs. NEW

go to the last Section : Section XXVI - Outline and Purpose for Part V

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07