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A195411 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 3,4,5 right triangle ABC.

%I A195411 #6 Mar 30 2012 18:57:45
%S A195411 6,2,9,7,8,7,2,0,2,2,0,0,9,1,5,1,1,5,5,8,4,3,1,7,8,2,0,2,0,7,6,2,4,2,
%T A195411 9,0,1,2,4,9,2,0,7,0,2,7,5,8,5,4,9,3,4,3,9,5,9,7,0,0,0,9,8,1,3,7,7,3,
%U A195411 1,1,3,4,1,0,7,7,9,5,2,1,2,0,6,7,3,6,7,6,7,3,0,7,0,5,6,2,8,1,4,3
%N A195411 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 3,4,5 right triangle ABC.
%C A195411 See A195304 for definitions and a general discussion.
%e A195411 Philo(ABC,G)=0.629787202200915115584317820207624290124920...
%t A195411 a = 3; b = 4; h = 2 a/3; k = b/3;
%t A195411 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195411 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195411 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195411 RealDigits[%, 10, 100]   (* (A) A195304 *)
%t A195411 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195411 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195411 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195411 RealDigits[%, 10, 100]   (* (B) A195305 *)
%t A195411 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195411 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195411 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195411 RealDigits[%, 10, 100]   (* (C) A195306 *)
%t A195411 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195411 RealDigits[%, 10, 100]   (* Philo(ABC,G) A195411 *)
%Y A195411 Cf. A195304.
%K A195411 nonn,cons
%O A195411 0,1
%A A195411 _Clark Kimberling_, Sep 18 2011