OFFSET
0,1
COMMENTS
A Soddyian triangle is a triangle whose outer Soddy circle has degenerated into a straight line. Its side lengths are related by the equation 1/sqrt(s-c)=1/sqrt(s-b)+1/sqrt(s-a) where the sides a<=b<=c and s is the semiperimeter. If the side lengths of such a triangle form an arithmetic progression 1, 1+d, 1+2d, where d is the common difference, then d = 0.5165877... and is the solution to the equation 37d^4+36d^3+6d^2-12d-3 = 0 such that 0<d<1. This triangle has angles of approx. 105.96, 45.82 and 28.22 degs.
LINKS
F. M. Jackson, Soddyian triangles, Forum Geometr. 13 (2013), 1-6.
FORMULA
d = (-18+16*sqrt(3)+37*sqrt((608*sqrt(3))/1369-240/1369))/74.
EXAMPLE
0.51658772215405264712532988077485052478638588883477756993492758314966...
MATHEMATICA
a=1; b=1+d; c=1+2d; s=(a+b+c)/2; sol=Solve[1/Sqrt[s-a]+1/Sqrt[s-b]-1/Sqrt[s-c]==0&&0<d<1, d]; RealDigits[N[d /. sol[[1]], 100]][[1]]
PROG
(PARI) polrootsreal(37*x^4+36*x^3+6*x^2-12*x-3)[2] \\ Charles R Greathouse IV, Apr 16 2014
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Frank M Jackson, Aug 23 2013
STATUS
approved