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 X - Analysis for Region One

Summary for the Section :

• In this Section , we shall take a closer look at Region One ,
  • developing the necessary relations for our later use .

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Setting up length [ RHO ] and angle [ THETA ] :

• Let us now set up the length variable [ RHO ] and the angle [ THETA ] ,
  • which then fully define the position of [ Point H ] ; same as before .

  

  The co-ordinates for [ Point H ] is then given by :

  • 

  • 

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Setting up the temporary angles PHI / PSI / GAMMA :

• Let us now set up the 3 temporary angles [ PHI ] , [ PSI ] and [ GAMMA ] based on [ THETA ] :
  • for our use in this Section only .

  

  The assigned values for the 3 angles here are then :

  • 

  • 

  • 

• We can then set up the following relations arising therefrom :
  • 

  • 

  • 

    for our use below .

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Calculating the [ Length of H-A ] :

• Let us now take a quick look at the position of [ Point A ] .

  

  We then have , based on the diagram on-the-left above , this relation for the [ Length of H-A ] :

  • 

  Consequently :

  • 

• Let us bring-in this relation from above :
  • 

  Substituting this into the equation immediately above then yields us this :

  • 

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Calculating the [ Length of H-B ] :

• Let us now take a quick look at the position of [ Point B ] .

  

  We then have , based on the diagram on-the-left above , this relation for the [ Length of H-B ] :

  • 

  Consequently :

  • 

• Let us bring-in this relation from above :
  • 

  Substituting this into the equation immediately above then yields us this :

  • 

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Calculating the [ Length of H-C ] :

• Let us now take a quick look at the position of [ Point C ] .

  

  We then have , based on the diagram on-the-left above , this relation for the [ Length of H-C ] :

  • 

  Consequently :

  • 

• Let us bring-in this relation from above :
  • 

  Substituting this into the equation immediately above then yields us this :

  • 

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Summarizing :

• We have therefore established these 3 relations for Region One :
  • 
  • 
  • 

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go to the next Section : Section XI - Analysis for Region Two

go to the last Section : Section IX - Setting up the 3 Key Variables [ R ] / [ Rho ] / [ Theta ]

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07