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A195486 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the sqrt(2),sqrt(5),sqrt(7) right triangle ABC.

%I A195486 #5 Mar 30 2012 18:57:47
%S A195486 6,2,0,3,3,2,2,7,2,6,5,3,0,2,5,8,3,8,0,5,5,6,8,6,8,3,7,2,0,6,0,7,6,8,
%T A195486 8,6,4,8,3,6,1,3,4,8,2,5,4,2,4,8,1,9,1,4,6,1,8,9,3,2,4,2,5,0,2,3,7,3,
%U A195486 1,4,7,9,0,4,8,7,0,3,3,4,1,5,9,1,5,2,4,7,6,8,7,4,2,0,1,3,7,2,0,9
%N A195486 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the sqrt(2),sqrt(5),sqrt(7) right triangle ABC.
%C A195486 See A195304 for definitions and a general discussion.
%e A195486 Philo(ABC,G)=.62033227265302583805568683720607688648361...
%t A195486 a = Sqrt[2]; b = Sqrt[5]; h = 2 a/3; k = b/3;
%t A195486 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195486 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195486 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195486 RealDigits[%, 10, 100] (* (A) A195483 *)
%t A195486 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195486 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195486 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195486 RealDigits[%, 10, 100] (* (B) A195484 *)
%t A195486 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195486 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195486 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195486 RealDigits[%, 10, 100] (* (C) A195485 *)
%t A195486 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195486 RealDigits[%, 10, 100] (* Philo(ABC,G) A195486 *)
%Y A195486 Cf. A195304.
%K A195486 nonn,cons
%O A195486 0,1
%A A195486 _Clark Kimberling_, Sep 19 2011