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A195478 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,sqrt(3),2 right triangle ABC.

%I A195478 #5 Mar 30 2012 18:57:47
%S A195478 6,1,3,8,4,1,7,2,5,3,9,4,1,8,6,8,1,0,6,6,0,3,6,7,2,4,6,0,0,1,3,9,4,0,
%T A195478 2,6,6,0,7,4,8,2,7,9,6,4,8,4,2,3,9,2,9,9,9,3,8,3,0,9,0,1,7,7,7,0,9,5,
%U A195478 7,8,7,7,1,4,1,7,5,6,4,4,4,3,6,8,4,1,2,8,9,0,4,7,2,2,2,1,4,2,9,1
%N A195478 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,sqrt(3),2 right triangle ABC.
%C A195478 See A195304 for definitions and a general discussion.
%e A195478 Philo(ABC,G)=0.61384172539418681066036724600139402660748...
%t A195478 a = 1; b = Sqrt[3]; h = 2 a/3; k = b/3;
%t A195478 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195478 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195478 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195478 RealDigits[%, 10, 100] (* (A) A195575 *)
%t A195478 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195478 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195478 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195478 RealDigits[%, 10, 100] (* (B) A195576 *)
%t A195478 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195478 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195478 f3 = (f[t])^(1/2) /. Part[s, 4]
%t A195478 RealDigits[%, 10, 100] (* (C) A195577 *)
%t A195478 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195478 RealDigits[%, 10, 100] (* Philo(ABC,G) A195578 *)
%Y A195478 Cf. A195304.
%K A195478 nonn,cons
%O A195478 0,1
%A A195478 _Clark Kimberling_, Sep 19 2011