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%I #28 Oct 27 2023 10:41:01
%S 5,1,6,5,8,7,7,2,2,1,5,4,0,5,2,6,4,7,1,2,5,3,2,9,8,8,0,7,7,4,8,5,0,5,
%T 2,4,7,8,6,3,8,5,8,8,8,8,3,4,7,7,7,5,6,9,9,3,4,9,2,7,5,8,3,1,4,9,6,6,
%U 2,6,7,5,5,1,9,2,9,4,5,0,4,9,8
%N Positive real root of 37*x^4+36*x^3+6*x^2-12*x-3.
%C 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.
%H F. M. Jackson, <a href="http://forumgeom.fau.edu/FG2013volume13/FG2013index.html">Soddyian triangles</a>, Forum Geometr. 13 (2013), 1-6.
%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%F d = (-18+16*sqrt(3)+37*sqrt((608*sqrt(3))/1369-240/1369))/74.
%e 0.51658772215405264712532988077485052478638588883477756993492758314966...
%t 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]]
%o (PARI) polrootsreal(37*x^4+36*x^3+6*x^2-12*x-3)[2] \\ _Charles R Greathouse IV_, Apr 16 2014
%Y Cf. A210484.
%K nonn,cons
%O 0,1
%A _Frank M Jackson_, Aug 23 2013
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