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A195490 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the sqrt(7),3,4 right triangle ABC.

%I A195490 #5 Mar 30 2012 18:57:47
%S A195490 6,3,5,0,0,3,8,3,3,3,3,6,2,3,7,3,5,2,4,7,0,2,1,2,1,9,0,3,6,9,3,5,0,3,
%T A195490 5,9,3,1,9,3,7,8,2,0,9,4,7,3,1,4,8,3,5,1,7,0,6,8,1,4,0,6,5,2,9,7,0,2,
%U A195490 5,4,4,1,6,5,9,8,5,1,3,1,3,7,7,1,4,9,2,3,0,8,8,2,4,9,0,9,4,6,4,0
%N A195490 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the sqrt(7),3,4 right triangle ABC.
%C A195490 See A195304 for definitions and a general discussion.
%e A195490 Philo(ABC,G)=0.635003833336237352470212190369350359...
%t A195490 a = Sqrt[7]; b = 3; h = 2 a/3; k = b/3;
%t A195490 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195490 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195490 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195490 RealDigits[%, 10, 100] (* (A) A195487 *)
%t A195490 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195490 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195490 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195490 RealDigits[%, 10, 100] (* (B) A195488 *)
%t A195490 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195490 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195490 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195490 RealDigits[%, 10, 100] (* (C) A195489  *)
%t A195490 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195490 RealDigits[%, 10, 100] (* Philo(ABC,G) A195490 *)
%Y A195490 Cf. A195304.
%K A195490 nonn,cons
%O A195490 0,1
%A A195490 _Clark Kimberling_, Sep 19 2011