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 XVII - Confirming the Existance of the 2 Equilateral Triangles

Summary for the Section :

• In this Section , we shall confirm the existance of the 2 Equilateral Triangles .

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Recalling the Position Vectors :

• Let us bring back from the last Section ,
  • this set of 3 expressions for the Position Vectors of [ Point A ] / [ Point B ] / [ Point C ] :
    • 

    • 

    • 

  As such ,

  • the X-Y co-ordinates of [ Point A ] is given by :
    • 

    • 

  • the X-Y co-ordinates of [ Point B ] is given by :
    • 

    • 

  • the X-Y co-ordinates of [ Point C ] is given by :
    • 

    • 

• Let us now re-write the above 6 expressions in this matrix format below :
  • segregating { the portion that varies with [ R ] } from { the portion that varies with [ Rho ] } .
    • 

    • 

    • 

  Consequently , we can further re-write the above 3 expressions

  • in this special format :
    • 

    • 

    • 

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Structural Considerations :

• Based on the above 3 Matrix Equations , we shall now set up this set of 6 vectors below :
  • for the { portion that varies with [ R ] }, we set up :
    • 

    • 

    • 

  • for the { portion that varies woth [ Rho ] } , we set up :
    • 

    • 

    • 

  so that :

  • we can now re-write the original 3 Matrix Equations in this format :
    • [ Vector O-A ] = [ Vector O-A' ] + [ Vector O-A" ]

    • [ Vector O-B ] = [ Vector O-B' ] + [ Vector O-B" ]

    • [ Vector O-C ] = [ Vector O-C' ] + [ Vector O-C" ]

      as per this set of 6 diagrams below .

  

  

• And we take note here that :
  • For the 3 vectors [ Vector O-A' ] / [ Vector O-B' ] / [ Vector O-C' ] :
    • the 3 vectors are of-the-same-size and point in 3 different directions ,
      • with each direction being different from another direction by an exact angle of [ 120 degrees ] , or [ pi-over-three ] .

    As such ,

    • { Triangle A'-B'-C' } is an equilateral triangle with [ Point O ] being its centroid .

  • For the 3 vectors [ Vector O-A" ] / [ Vector O-B" ] / [ Vector O-C" ] :
    • the 3 vectors are of-the-same-size and point in 3 different directions ,
      • with each direction being different from another direction by an exact angle of [ 120 degrees ] , or [ pi-over-three ] .

    As such ,

    • { Triangle A"-B"-C" } is an equilateral triangle with [ Point O ] being its centroid .

  

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The Centroid of { Triangle A-B-C } :

• Let us now bring back these 3 Equations from above :
  • [ Vector O-A ] = [ Vector O-A' ] + [ Vector O-A" ]

  • [ Vector O-B ] = [ Vector O-B' ] + [ Vector O-B" ]

  • [ Vector O-C ] = [ Vector O-C' ] + [ Vector O-C" ]

  Since [ Point O ] is the centroid of { Triangle A'-B'-C' } , we have :

  • [ Vector O-A' ] + [ Vector O-B' ] + [ Vector O-C' ] = [ zero ]

  Since [ Point O ] is the centroid of { Triangle A"-B"-C" } , we have :

  • [ Vector O-A" ] + [ Vector O-B" ] + [ Vector O-C" ] = [ zero ]

• And based on the above 2 equations , we can now write :
  • [ Vector O-A ] + [ Vector O-B ] + [ Vector O-C ] = [ zero ]

  As such , we can now conclude that [ Point O ] is indeed the centroid of { Triangle A-B-C } .

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Summarizing on Part III :

• Based on our findings so far , we can now make this statement :
  • For any { triangle } or { 3-Points Shape } constructed via our { Standardized Construction Procedure } here in Part III ,
    • we can always break-down the said { Triangle } or { 3-Points Shape } into a pair of mirror-imaged Equilateral Triangles ,
      • resized and/or re-oriented as may be appropriate .

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What's Next :

• The question then arises :
  • as to whether or not we would be able solve for the size and orientation of the 2 Equilateral Triangle ,
    • once we have a fix on original { Triangle A-B-C } or alternately the { 3-Points Shape } ,
      • as the case may be .

  And we shall look at this next , in Part IV of this paper to follow .

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

go to the last Section : Section XVI - Summarizing our Findings on the last 6 Sections

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07