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 III - The Centroid as the Center of the Triangle

Summary for the Section :

• In this Section , we shall take a first look at the pros and cons of :
  • using the Centroid of the Triangle as our primary Point-Of-Reference .

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Using the Centroid as the primary Point-Of-Reference :

• In general , for any given triangle { Triangle A-B-C } , the Centroid can usually be regarded as the center-point of the said triangle ,
  • as identified by [ Point O ] in the diagram in-the-middle below .

  

  And once we have identified [ Point O ] as the Centroid ,

  • we can then set up the 3 position-vectors :
    • [ Vector O-A ] , [ Vector O-B ] and [ Vector O-C ] , respectively ;

      as per the diagram on-the-right above ,

    to fully define { Triangle A-B-C } .

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Procedure to identify the Centroid :

• Let us now quickly review the procedure to identify the Centroid of { Triangle A-B-C } .

  First , we identify :

  • the mid-point of [ Segment A-B ] as [ point D ] ,

  • the mid-point of [ Segment B-C ] as [ point E ] , and

  • the mid-point of [ Segment C-A ] as [ point F ] ;

    using a { Straightedge-and-Compass Procedure } as shown below .

  

  We can then use a Straightedge to create :

  • [ Line-Seqment A-E ] , [ Line-Seqment B-F ] and [ Line-Segment C-D ] respectively ,
    • as shown in the diagram in-the-middle below .

  

  And the intersection of the 3 line-segments would then give us [ Point O ] ,

  • the Centroid of { Triangle A-B-C } .

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The Problem of Syzygy :

• Let us now take a quick look at what happens when :
  • [ Point A ] , [ Point B ] and [ Point C ] happen to lie on the same straight-line :

  

  In this case , the Centroid is not defined and apparently disappears .

• Our proposal here is therefore to use :
  • the [ Center-of-Mass ] for the { Three-Body Problem with 3 Equal Masses }
    • as the primary Point-Of-Reference for the { 3-Points Shape } formed by the 3 equal masses ,

      namely :

      • [ Mass A ] , [ Mass B ] , and [ Mass C ] respectively .

  And we shall take a quick look at this , next .

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go to the next Section : Section IV - The Center-of-Mass for the Three-Body Problem with 3 Equal Masses

go to the last Section : Section II - An Overview of the Paper

return to the Pair of Mirror-Imaged Equilateral Triangles Homepage

Original dated 2021-9-07