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A195449 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,3,sqrt(10) right triangle ABC.

%I A195449 #5 Mar 30 2012 18:57:45
%S A195449 5,6,1,7,0,8,1,6,9,7,8,3,3,4,4,5,9,5,1,7,8,9,1,5,7,7,2,9,4,0,4,7,3,9,
%T A195449 5,6,0,3,4,0,3,8,8,0,0,2,4,5,9,2,5,6,8,4,0,2,5,6,5,9,8,4,3,8,4,8,9,1,
%U A195449 0,8,5,4,3,8,7,7,7,9,4,2,7,1,9,7,7,1,3,4,8,3,9,8,9,9,0,9,4,1,4,8
%N A195449 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 1,3,sqrt(10) right triangle ABC.
%C A195449 See A195304 for definitions and a general discussion.
%e A195449 Philo(ABC,G)=0.561708169783344595178915772940473956034038800...
%t A195449 a = 1; b = 3; h = 2 a/3; k = b/3;
%t A195449 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195449 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195449 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195449 RealDigits[%, 10, 100] (* (A) A195446  *)
%t A195449 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195449 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195449 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195449 RealDigits[%, 10, 100] (* (B) A195447 *)
%t A195449 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195449 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195449 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195449 RealDigits[%, 10, 100] (* (C) A195448 *)
%t A195449 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195449 RealDigits[%, 10, 100] (* Philo(ABC,G) A195449 *)
%Y A195449 Cf. A195304.
%K A195449 nonn,cons
%O A195449 0,1
%A A195449 _Clark Kimberling_, Sep 18 2011