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A243508
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Decimal expansion of the real positive root of 48x^4 + 16x^3 - 27x^2 - 18x - 3 = 0.
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0
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8, 8, 0, 5, 8, 3, 3, 4, 8, 3, 3, 9, 8, 2, 8, 1, 2, 4, 2, 1, 2, 9, 2, 3, 7, 8, 3, 7, 8, 4, 4, 9, 8, 7, 4, 3, 6, 8, 2, 4, 1, 8, 6, 4, 8, 4, 6, 8, 1, 5, 3, 1, 7, 1, 8, 1, 1, 0, 0, 1, 8, 1, 8, 6, 8, 5, 4, 4, 8, 4, 7, 7, 0, 5, 6, 8, 1, 6, 5, 2, 8, 3, 6, 5, 2
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OFFSET
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0,1
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COMMENTS
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Given an equilateral triangle of unit length with two cevians drawn from one vertex to the side opposite that divide the equilateral triangle into 3 subtriangles. Adjust these cevians so that the 3 subtriangles all have congruent incircles. Then the real positive root of 48x^4 + 16x^3 - 27x^2 - 18x - 3 = 0 gives x = 1/4(3^(1/3)+3^(2/3)) = 0.880583348... as the length of these cevians and the radius of the three congruent incircles is given by A/(s+2x) where A is the area and s the semiperimeter of the equilateral triangle. Hence the congruent inradius = Sqrt(3)/(2(3+3^(2/3)+3^(1/3)).
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LINKS
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FORMULA
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48x^4 + 16x^3 - 27x^2 - 18x - 3 has real positive root x = 1/4(3^(1/3)+3^(2/3)) = 0.880583348...
16*x^3 - 9*x - 3 is the irreducible polynomial. - Michael Somos, Jun 09 2014
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MATHEMATICA
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N[Select[x/.Solve[48x^4+16x^3-27x^2-18x-3==0, {x}], Im[#]==0&&Re[#]>0 &], 100]
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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