OFFSET
0,1
COMMENTS
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.
EXAMPLE
0.3778323013337351599409973840073962246062292271031367462170448060876665371704...
MATHEMATICA
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]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Frank M Jackson, Aug 04 2011
STATUS
approved