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 VIII - Some Operational Features of our Construction System

Summary for the Section :

• In this Section , we shall take a quick look at :
  • some of the operational features of our Construction System for { Triangle A-B-C } .

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Splitting the Reference Structure into 4 Zones :

• For a better understanding on how our { Construction System } for { Triangle A-B-C } works ,
  • let us now`break-down our { Reference Structure } for our { Standardized Construction Procedure } into 4 different zones ,
    • as follows :
      • the { Inner Region } being the area covered by the inscribed equilateral triangle { Triangle D-E-F } ,

        and

      • the outer 3 regions being { Region Aaron } , { Region Baron } and { Region Cyron } respectively ,

        as shown in the diagram on-the-left below .

  

  And we take note at this point that :

  • When [ Point H ] happens to land in { Region Aaron } ,
    • [ Point A ] would be located in the [ Inner Region ] ;

  • When [ Point H ] happens to land in { Region Baron } ,
    • [ Point B ] would be located in the [ Inner Region ] ;

  • When [ Point H ] happens to land in { Region Cyron } ,
    • [ Point C ] would be located in the [ Inner Region ] ;

    as per the diagrams on the middle-left , the middle-right and the very-right above .

• And by the same token :
  • When [ Point H ] happens to land on [ Line D-E ] ,
    • [ Point A ] and [ Point H ] will occupy the same point ,
      • because both [ distance H-T ] and [ distance A-T ] will be [ zero ] ;

  • When [ Point H ] happens to land on [ Line E-F ] ,
    • [ Point B ] and [ Point H ] will occupy the same point ,
      • because both [ distance H-U ] and [ distance B-U ] will be [ zero ] ;

  • When [ Point H ] happens to land on [ Line F-D ] ,
    • [ Point C ] and [ Point H ] will occupy the same point ,
      • because both [ distance H-V ] and [ distance A-V ] will be [ zero ] ;

    as per this set of 12 diagrams below .

  

  

  

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Special Cases of Double-Collision vs. the Equilateral Triangle :

• We take note here that :
  • When [ Point H ] happens to lie upon [ Point D ] ,
    • [ Point C ] and [ Point A ] will also lie upon [ Point D ] creating a double-collision situation ;

  • When [ Point H ] happens to lie upon [ Point E ] ,
    • [ Point A ] and [ Point B ] will also lie upon [ Point E ] creating a double-collision situation ;

  • When [ Point H ] happens to lie upon [ Point F ] ,
    • [ Point B ] and [ Point C ] will also lie upon [ Point F ] creating a double-collision situation ;

    as per the 3 diagrams on the very-left , the middle-left , and the middle-right below .

  

  And in contrast :

  • when [ Point H ] happens to lie at the [ center of the Circle ] ,
    • the resultant { Triangle A-B-C } is an equilateral triangle ;
      • as per the diagram on the very-right above .

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The Cases of Syzygy :

Let us now take a look at what happens when [ Point H ] happens to lie upon the edge of the circle ;
• as per this set of 12 diagrams below .







And we simply take note here that :

• when [ Point H ] happens to land on the [ edge of the circle ] ,
  • [ Point A ] , [ Point B ] and [ Point C ] will always be in the formation of a straight-line ,
    • thereby creating the case of syzygy .

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The Isosceles Triangles :

• For our demonstration to follow immediately below ,
  • let us now set up 3 more points on the edge of the circle , namely :
    • [ Point D' ] , [ Point E' ] and [ Point F' ] respectively ,

    so that :

    • [ Line D-D' ] is perpendicular to [ Line E-F ] ,

    • [ Line E-E' ] is perpendicular to [ Line F-D ] ,

    • [ Line F-F' ] is perpendicular to [ Line D-E ] ;

      as per this set of 4 diagrams below .

  

  And we take note here that :

  • When [ Point H ] happens to land on [ Line D-D' ] ,
    • the resultant { Triangle A-B-C } is an isosceles triangle ;

  

  • When [ Point H ] happens to land on [ Line E-E' ] ,
    • the resultant { Triangle A-B-C } is an isosceles triangle ;

  

  • When [ Point H ] happens to land on [ Line F-F' ] ,
    • the resultant { Triangle A-B-C } is an isosceles triangle ;

  

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What's Next :

• In the next 9 Sections , we shall attempt to establish that :
  • all triangles can always be broken down into
    • a pair of mirror-imaged equilateral triangles ,
      • resized and/or re-oriented as the case may be .

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

go to the last Section : Section VII - Constructing { Triangle A-B-C }

return to Pair of Mirrored-Imaged Equilateral Triangles HomePage

Original dated 2021-9-07