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A176269
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Decimal expansion of the value of d that maximizes the area of a triangle whose angles form a harmonic progression in the ratio 1 : 1/(1-d) : 1/(1-2d), whose side length opposite the largest or smallest angle remains constant and for d in the interval [-oo, 1/2] where 1/(1-d) and 1/(1-2d) are always positive.
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0
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3, 7, 7, 8, 3, 2, 3, 0, 1, 3, 3, 3, 7, 3, 5, 1, 5, 9, 9, 4, 0, 9, 9, 7, 3, 8, 4, 0, 0, 7, 3, 9, 6, 2, 2, 4, 6, 0, 6, 2, 2, 9, 2, 2, 7, 1, 0, 3, 1, 3, 6, 7, 4, 6, 2, 1, 7, 0, 4, 4, 8, 0, 6, 0, 8, 7, 6, 6, 6, 5, 3, 7, 1, 7, 0, 4, 8, 3, 3, 5, 5, 3, 6, 4, 9, 0, 5, 8, 4, 4, 7, 0, 5, 6, 7, 2, 7, 3, 1, 3, 5, 7
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OFFSET
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0,1
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COMMENTS
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If the angles of a triangle form a harmonic progression in the ratio 1 : 1/(1-d) : 1/(1-2d) where d is the common difference between the denominators of the harmonic progression, then the angles are expressed as
Pi*(1-d)*(1-2d)/(2d^2-6d+3), Pi*(1-2d)/(2d^2-6d+3), Pi*(1-d)/(2d^2-6d+3).
When d = 0.3778323013... it uniquely maximizes the area of this triangle for a constant side length opposite the largest or smallest angle. This triangle has approximate internal angles 26.866 degrees, 43.181 degrees, 109.953 degrees.
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LINKS
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EXAMPLE
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0.3778323013337351599409973840073962246062292271031367462170448060876665371704...
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MATHEMATICA
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NMaximize[{Sqrt[(Sin[Pi*(1-d)*(1-2d)/(2d^2-6d+3)]+Sin[Pi*(1-2d)/(2d^2-6d+3)]+Sin[Pi*(1-d)/(2d^2-6d+3)])*(-Sin[Pi*(1-d)*(1-2d)/(2d^2-6d+3)]+Sin[Pi*(1-2d)/(2d^2-6d+3)]+Sin[Pi*(1-d)/(2d^2-6d+3)])*(Sin[Pi*(1-d)*(1-2d)/(2d^2-6d+3)]-Sin[Pi*(1-2d)/(2d^2-6d+3)]+Sin[Pi*(1-d)/(2d^2-6d+3)])*(Sin[Pi*(1-d)*(1-2d)/(2d^2-6d+3)]+Sin[Pi*(1-2d)/(2d^2-6d+3)]-Sin[Pi*(1-d)/(2d^2-6d+3)])]/(4*Sin[Pi*(1-d)(1-2d)/(2d^2-6d+3)]^2), d<1/2}, d, AccuracyGoal->120, PrecisionGoal->100, WorkingPrecision->240]
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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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