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 IX - Setting up the 3 Key Variables [ R ] / [ Rho ] / [ Theta ]

Summary for the Section :

• In this Section , we shall set up the 3 key variables , namely :
  • [ R ] , [ RHO ] , and [ THETA ] ;

    for our use in the next 8 Sections .

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Setting up [ R ] / [ RHO ] / [ THETA ] :

• Let us bring in again the { Reference Structure } for our Construction System , namely :
  • an equilateral triangle inscribed in a { Circle of radius R } .
    • as per the diagram on-the-right below .

  And [ radiuse R ] is the first key variable [ R ] we shall be using throughout the next 8 Sections .

  

• And for our analysis to follow in the next 8 Sections ,
  • it is necessary for us to put in a set of { 2 orthogonal axis } for referencing the { plane of the Triangle } .

  And we shall do so now by :

  • setting up a { X-Axis } orthogonal to { Line D-E } passing thru the center-of-the circle [ Point O ]

    and correspondingly :

  • setting up a { Y-Axis } orthogonal to the { X-Axis } also passing thru [ Point O ] ;
    • as per the diagram in-the-middle above .

• Let us now also set up :
  • the length variable [ RHO ] and the angle variable [ THETA ] for the position vector [ Vector O-H ] ,

  so that :

  • [ RHO ] is the length of position vector [ Vector O-H ] ,

    and

  • [ THETA ] is the angle that the position vector [ Vector O-H ] makes with the { X-Axis } ;
    • as per the diagram on-the-right above .

  Consequently , the co-ordinates for [ Point H ] is given by :

  • 

  • 

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Splitting the Circle Area into 6 Regions :

• In the next 6 Sections, we shall attempt to express :
  • the length of [ Line Segment H-A ] ,

  • the length of [ Line Segment H-B ] , and

  • the length of [ Line Segment H-C ] ,

    as functions of [ R ] , [ RHO ] and [ THETA ] .

  

  And for the ease of analysis ,

  • we shall now break down the { Reference Structure } into 6 regions ,
    • as shown below .

  <

• And we shall be able to establish , by the end of the next 6 Sections , that :
  • 

  • 

  • 

    for all 6 regions ;

  so that :

  • there will not be any boundary problems as we move across from one Region to another Region .

  Additionally , we also take special note here that :

  

  • when [ Point H ] happen to lie in { Region Aaron } ,
    • the 1st formula above for [ L-HA ] will yield a negative value ;

  • when [ Point H ] happen to lie in { Region Baron } ,
    • the 2nd formula above for [ L-HB ] will yield a negative value ;

  • when [ Point H ] happen to lie in { Region Cyron } ,
    • the 3rd formula above for [ L-HC ] will also yield a negative value .

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What's next :

• In the next 6 Sections , we shall look at each of the 6 Regions separately :
  • to develop the fomulas for the [ length-of-HA ] , [ length-of-HB ] and [ length-of-HC ] .

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go to the next Section : Section X - Analysis for Region One

go to the last Section : Section VIII - Some Operational Features of our Construction System

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07