OFFSET
1,2
COMMENTS
If the sides of a triangle form a geometric progression in the ratio 1:r:r^2 then when r=1.378874314... it uniquely maximizes the area of the triangle. This triangle has approximate internal angles of 30.534, 44.470 and 104.996 degrees.
FORMULA
Sqrt(1/2 + (4-sqrt(921)/9)^(1/3)/2 + (4+sqrt(307/27))^(1/3)/2). - Charles R Greathouse IV, Feb 13 2011
EXAMPLE
1.378874314...
MATHEMATICA
N[Maximize[Sqrt[(1 + r + r^2)*(1 + r - r^2)*(1 - r + r^2)*(-1 + r + r^2)]/4, r], 100]
Root[ 2#^6-3#^4-#^2-1 &, 1] // RealDigits[#, 10, 100]& // First (* Jean-François Alcover, Mar 05 2013 *)
PROG
(PARI) solve(r=1, 2, 2*r^6 - 3*r^4 - r^2 - 1) \\ Charles R Greathouse IV, Feb 13 2011
CROSSREFS
KEYWORD
AUTHOR
Frank M Jackson, Nov 13 2010
EXTENSIONS
Edited by Frank M Jackson, Nov 13 2010
STATUS
approved