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A195424 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 5,12,13 right triangle ABC.

%I A195424 #6 Mar 30 2012 18:57:45
%S A195424 5,8,4,7,1,3,2,5,5,9,5,0,1,4,2,2,4,5,6,7,6,1,2,1,4,1,6,4,2,7,0,6,2,1,
%T A195424 7,4,5,9,1,6,1,6,2,7,0,2,9,2,0,4,7,0,4,5,4,7,8,0,8,9,1,5,8,3,6,9,1,8,
%U A195424 0,5,9,4,1,7,7,6,9,8,5,7,4,3,1,8,8,1,4,1,4,4,1,1,7,3,8,7,4,9,9,9
%N A195424 Decimal expansion of normalized Philo sum, Philo(ABC,G), where G=centroid of the 5,12,13 right triangle ABC.
%C A195424 See A195304 for definitions and a general discussion.
%e A195424 Philo(ABC,G)=0.584713255950142245676121416427062174591616270...
%t A195424 a = 5; b = 12; h = 2 a/3; k = b/3;
%t A195424 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195424 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195424 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195424 RealDigits[%, 10, 100] (* (A) A195412 *)
%t A195424 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195424 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195424 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195424 RealDigits[%, 10, 100] (* (B) A195413 *)
%t A195424 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195424 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195424 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195424 RealDigits[%, 10, 100] (* (C) A195414 *)
%t A195424 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195424 RealDigits[%, 10, 100] (* Philo(ABC,G) A195424 *)
%Y A195424 Cf. A195304.
%K A195424 nonn,cons
%O A195424 0,1
%A A195424 _Clark Kimberling_, Sep 18 2011