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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
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