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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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%I #29 Jul 01 2023 16:55:50

%S 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,

%T 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,

%U 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

%N 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.

%C 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

%C Pi*(1-d)*(1-2d)/(2d^2-6d+3), Pi*(1-2d)/(2d^2-6d+3), Pi*(1-d)/(2d^2-6d+3).

%C 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.

%e 0.3778323013337351599409973840073962246062292271031367462170448060876665371704...

%t 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]

%K nonn,cons

%O 0,1

%A _Frank M Jackson_, Aug 04 2011