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 XIX - Setting up the Random Triangle and the Reference Frame

Summary for the Section :

• In this Section , we shall set up the random triangle { Triangle A-B-C } and its Reference Frame ,
  • for our use throughout this Part IV of the paper .

Setting up the general triangle { Triangle A-B-C } :

• For our discussion purposes throughout this Part IV of the paper ,
  • we shall first set up three (3) random points on a { 2-Dimensional Plane } ,
    • namely [ Point A ] , [ Point B ] and [ Point C ] respectively ;

    so that :

    • the 3 points are separate from one-another and do not lie on the same straight-line .

  As such , the { 3-points shape } formed by [ Point A ] , [ Point B ] and [ Point C ] is a triangle ,

  • and this { 3-points shape } shall be referred to as { Triangle A-B-C } throughout this Part IV of the paper .

  

• We can then identify [ Point O ] as the centroid of { Triangle A-B-C } ,
  • using the procedure we have previous discussed in Section III of this paper ;

  and

  • set up the 3 position vectors [ Vector O-A ] , [ Vector O-B ] , and [ Vector O-C ] respectively ;
    • as per the set of 3 diagrams above .

• And because [ Point O ] is the centroid of { Triangle A-B-C } , we always have :
  • [ Vector O-A ] + [ Vector O-B ] + [ Vector O-C ] = [ zero ]

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Setting up the Reference Frame for the { 2-Dimensional Plane } :

• For our analysis to follow , it is necessary for us to set up a Reference Frame and quantify the distances .

  And we shall do so now :

  • FIRST ,
    • we set up a random straight-line passing thru [ Point O ] to act as the [ X-Axis ] ;

  • SECONDLY ,
    • we set up another straight-line passing thru [ Point O ] orthogonal to the [ X-Axis ] to act as the [ Y-Axis ] ;

    as per the diagram on-the-left below .

  

• We can then draw in a circle of random radius , using [ Point O ] as the center of the circle ,
  • to act as our { unit-circle } for referencing distances ;

  so that :

  • the [ distance O-A ] , [ distance O-B ] , and [ distance O-C ] can now be quantified accordingly ;
    • as per the 2 diagrams in-the-middle and on-the-left above .

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The Position Vectors [ Vector O-A ] / [ Vector O-B ] / [ Vector O-C ] :

• Once we have set up the Reference Frame and quantified the distances ,
  • we can then set up the fixed-values that will fully define the 3 { position vectors } :
    • For [ Vector O-A ] :
      • we set up the distance [ R-sub-A ] and angle [ ALPHA ] ;

    • For [ Vector O-B ] :
      • we set up the distance [ R-sub-B ] and angle [ BETA ] ;

    • For [ Vector O-C ] :
      • we set up the distance [ R-sub-C ] and angle [ GAMMA ] ;

      as per this set of 3 diagrms below .

  

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What's Next :

• And with this in hand , we shall now move on to solve for the 2 Equilateral Triangles , next .

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go to the next Section : Section XX - Creating the 3rd-Order Polynomial Equation from { Triangle A-B-C }

go to the last Section : Section XVIII - Outline and Purpose for Part IV

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07