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 VII - Constructing { Triangle A-B-C }

Summary for the Section :

• In this Section , we shall construct the general { Triangle A-B-C } or alternately the { 3-Points Shape A-B-C in Syzygy } via :
  • First :
    • setting up the { Reference Structure } for our construction purposes ;

  • Secondly :
    • selecting a seed point [ Point H ] within the { Reference Structure } ;

  • Thirdly :
    • constructing { Triangle A-B-C } based on [ Point H ] using our { Standardized Construction Procedure } .

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Setting up the { Reference Structure } :

• In our process to construct the general triangle { Triangle A-B-C } ,
  • it is necessary for us to bring-in a [ Reference Structure ] , namely :
    • an equilateral triangle inscribed in a circle ;
      • as per the diagram on-the-right below .

  

  And we take note here that this [ Reference Structure ] can be created via a { Straightedge-and-Compass } procedure , as follows :

  • STEP ONE :
    • We take out a piece of plain paper and mark-off a point to serve as the center for a circle ,
      • and we draw in a circle of [ radius R ] ; as per the diagram on-the-left above .

  • STEP TWO :
    • We use a straightedge to draw in a straight-line passing thru the center of the circle ,
      • intersecting the circle at two (2) points ; as per the diagram on-the-middle-left above .

  • STEP THREE :
    • We draw in 2 arcs of [ radius R ] using the 2 intersection points in STEP TWO as the centers for the 2 arcs ;
      • so that :
        • the circle is now divided into 6 equal [ arc sections ] ; as per the diagram on-the-middle-right above .

  • STEP FOUR :
    • We then mark-off the inscribed { Equilateral Triangle D-E-F }
      • using alternate points of the six (6) points on the circumference ;
        • as per the diagram on-the-right above .

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Selecting a seed-point [ Point H ] :

• In our procedure to construct { Triangle A-B-C } ,
  • it is necessary for us to select a seed point , namely [ Point H ] ,
    • being any point within the circle area , inclusive of :
      • the [ center of the circle ] and [ any point on the circumference ] .

  

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Constructing { Triangle A-B-C } :

• Once we have selected the seed point [ Point H ] ,
  • we can then proceed to construct { Triangle A-B-C } using our { Standardized Construction Procedure } .

  And our { Standardized Construction Procedure } is as follows :

  • STEP ONE :
    • We construct :
      • a straight-line passing thru [ Point H ] perpendicular to [ Line D-E ] ,
        • intersecting the said [ Line D-E ] at [ Point T ] ;

      • a straight-line passing thru [ Point H ] perpendicular to [ Line E-F ] ,
        • intersecting the said [ Line E-F ] at [ Point U ] ;

      • a straight-line passing thru [ Point H ] perpendicular to [ Line F-D ] ,
        • intersecting the said [ Line F-D ] at [ Point V ] ;

        as per this set of 3 diagrams below .

  

  • STEP TWO :
    • We can then :
      • mark-off [ Point A ] on the line perpendicular to [ Line D-E ] so that :
        • [ distance H-T ] and [ distance A-T ] are equal in length ;

      • mark-off [ Point B ] on the line perpendicular to [ Line E-F ] so that :
        • [ distance H-U ] and [ distance B-U ] are equal in length ;

      • mark-off [ Point C ] on the line perpendicular to [ Line F-D ] so that :
        • [ distance H-V ] and [ distance C-V ] are equal in length ;

        as per this set of 3 diagrams below .

  

  • STEP THREE :
    • We can then use a straightedge to join-up [ Line A-B ] , [ Line B-C ] and [ Line C-A ] to complete the triangle { Triangle A-B-C } ,
      • as per this set of 3 diagrams below .

  

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Construction using a { Straightedge-and-Compass Procedure } :

• And we take note here that :
  • once the position of [ Point H ] is determined ,
    • the positions of [ Point A ] , [ Point B ] and [ Point C ] can be readily obtainable
      • via a { straightedge-and-compass procedure } as per this set of 9 diagrams below .

  

  

  

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What's Next :

• Let us now take a quick look at some of the operational features of our { Standardized Construction Procedure } , next .

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

go to the last Section : Section VI - Outline and Purpose for Part III

return to Pair of Mirrored-Imaged Equilateral Triangles HomePage

Original dated 2021-9-07