triangles.txt

https://twitter.com/Cshearer41/status/1103717536921800704

Let a, b, c be the triangles, smallest to largest.

Observations from the illustration about the relative triangle sizes:
  
    side b = 2 * height a                   [observation ab]
    side c = 2 * height b                   [observation bc]

Let w be the rectangle's width and let h be its height.

Observations about the rectangle's relationship to the triangles:

    w = side c                              [observation w]
    h = height a + height c                 [observation h]

Let k = (sqrt 3) / 4

Facts about any equilateral triangle t:

    area t = k * (side t)^2                 [area fact]
    height t = 2k * side t                  [height fact]

Obtaining an expression for (side b) in terms of (side a):

    side b = 2 * height a                   Restatement of [observation ab]
           = 2 * (2k * side a)              Substituting [height fact]
           = 4k * side a                    [lemma ab]

Obtaining an expression for (side c) in terms of (side b):

    side c = 2 * height b                   Restatement of [observation bc]
           = 2 * (2k * side b)              Substituting [height fact]
           = 4k * side b                    [lemma bc]

Obtaining an expression for (side c) in terms of (side a):

    side c = 4k * side b                    Restatement of [lemma bc]
           = 4k * (4k * side a)             Substituting [lemma ab]
           = 16 * k^2 * side a              [lemma ac]

Obtaining an expression for the total area of the triangles in terms of (side a):

    total triangle area
        = (area a)
        + (2 * area b)
        + (area c)

        =  k * (side a)^2                   Substituting [area fact] 3 times
        + 2k * (side b)^2
        +  k * (side c)^2

        =  k * ( side a            )^2
        + 2k * ( 4k * side a       )^2      Substituting [lemma ab]
        +  k * ( 16 * k^2 * side a )^2      Substituting [lemma ac]

        =       k   * (side a)^2
        +  32 * k^3 * (side a)^2
        + 256 * k^5 * (side a)^2
      
        = (k + (32 * k^3) + (256 * k^5))    [triangle area lemma]
        * (side a)^2

Obtaining an expression for the height of the rectangle in terms of (side a):

    h = (height a) + (height c)             Restatement of [observation h]

      = (2 * k * side a)                    Substituting [height fact] 2 times
      + (2 * k * side c)

      = (2 * k * side a)
      + (2 * k * (16 * k^2 * side a))       Substituting [lemma ac]

      = (2k * side a)
      + (32 * k^3 * side a)
      
      = (2k + (32 * k^3)) * side a          [rectangle height lemma]

Obtaining an expression for the area of the rectangle in terms of (side a):

    rectangle area
        = w * h

        = side c                          Substituting [observation w]
        * (2k + (32 * k^3)) * side a      Substituting [rectangle height lemma]

        = 16 * k^2 * side a               Substituting [lemma ac]
        * (2k + (32 * k^3)) * side a

        = 16 * k^2 * (2k + (32 * k^3)) * (side a)^2
        
        = ( (32 * k^3) + (512 * k^5) ) * (side a)^2      [rectangle area lemma]

answer
    = total triangle area
    / rectangle area
    
    = ( ( k + (32 * k^3) + (256 * k^5) ) * (side a)^2 )   Substituting [triangle area lemma]
    / ( (     (32 * k^3) + (512 * k^5) ) * (side a)^2 )   Substituting [rectangle area lemma]
    
    = ( k + (32 * k^3) + (256 * k^5) )           Simplify fraction
    / (     (32 * k^3) + (512 * k^5) )
    
    = 2/3                                        Wolfram Alpha