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%I #18 Jul 01 2023 15:50:22
%S 2,1,5,8,5,6,1,2,2,6,1,1,5,1,8,2,0,1,1,8,1,4,9,7,1,0,5,7,1,6,2,7,8,3,
%T 0,3,2,9,7,7,4,0,9,9,4,8,6,4,4,6,5,1,7,2,4,4,4,0,3,0,9,4,4,9,5,8,7,1,
%U 6,9,0,4,3,0,1,6,5,7,3,5,8,7,8,5,3,0,5,5,7,5,3,8,7,2,4,9,5,9,1,1
%N Decimal expansion of the value of d that maximizes the expression (1+1/(1-d)+1/(1-2d))*(-1+1/(1-d)+1/(1-2d))*(1-1/(1-d)+1/(1-2d))*(1+1/(1-d)-1/(1-2d)) for d in the interval [-oo, 1/2] where 1/(1-d) and 1/(1-2d) are always positive.
%C If the sides of a triangle form a harmonic progression in the ratio 1 : 1/(1-d) : 1/(1-2d) then when d = 0.2158561226... it uniquely maximizes the area of the triangle. This triangle has approximate internal angles of 33.956, 45.423 and 100.621 degrees.
%e 0.2158561226...
%t NMaximize[{Sqrt[(1+1/(1-d)+1/(1-2d))(-1+1/(1-d)+1/(1-2d))(1-1/(1-d)+1/(1-2d))(1+1/(1-d)-1/(1-2d))]/4, 0<d<(25/100)}, d, AccuracyGoal->120, PrecisionGoal->100, WorkingPrecision->240]
%K nonn,cons,easy
%O 0,1
%A _Frank M Jackson_, Jul 17 2011
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