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# chris-martin/triangles.txt Created Mar 25, 2019

 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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