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A195482 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 2,sqrt(5),3 right triangle ABC.

%I A195482 #5 Mar 30 2012 18:57:47
%S A195482 6,3,5,2,6,8,6,0,4,8,3,9,3,3,6,2,1,8,8,1,1,5,0,6,2,7,8,2,7,6,4,4,5,8,
%T A195482 5,2,0,1,9,8,1,8,7,6,3,7,9,6,2,3,1,6,4,1,6,6,5,4,9,0,3,9,5,0,9,2,3,2,
%U A195482 4,9,7,4,7,8,4,8,9,0,1,9,2,5,2,7,0,1,9,9,5,0,1,6,9,5,6,6,2,1,2,8
%N A195482 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 2,sqrt(5),3 right triangle ABC.
%C A195482 See A195304 for definitions and a general discussion.
%e A195482 Philo(ABC,G)=0.635268604839336218811506278276445852019818763...
%t A195482 a = 2; b = Sqrt[5]; h = 2 a/3; k = b/3;
%t A195482 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195482 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195482 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195482 RealDigits[%, 10, 100] (* (A) A195479 *)
%t A195482 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195482 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195482 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195482 RealDigits[%, 10, 100] (* (B) A195480 *)
%t A195482 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195482 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195482 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195482 RealDigits[%, 10, 100] (* (C) A195481 *)
%t A195482 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195482 RealDigits[%, 10, 100] (* Philo(ABC,G) A195482 *)
%Y A195482 Cf. A195304.
%K A195482 nonn,cons
%O A195482 0,1
%A A195482 _Clark Kimberling_, Sep 19 2011