%I #11 Mar 30 2012 18:57:45

%S 4,8,4,7,8,2,3,8,5,3,6,6,1,7,5,3,4,8,3,3,5,1,6,5,4,1,8,0,2,2,8,1,1,5,

%T 2,7,8,0,8,8,2,5,5,4,5,2,2,8,2,5,9,9,2,3,4,1,2,9,5,4,4,3,3,4,6,0,2,1,

%U 8,8,6,9,4,6,2,9,6,2,9,3,6,8,4,9,2,7,9,5,9,9,8,0,7,0,1,2,2,0,6,2

%N Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 3,4,5 right triangle ABC.

%C See A195284 for definitions and a general discussion.

%e Philo(ABC,I)=0.4847823853661753483351654180...

%t a = 5; b = 12; c = 13;

%t h = a (a + c)/(a + b + c); k = a*b/(a + b + c);

%t f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2;

%t s = NSolve[D[f[t], t] == 0, t, 150]

%t f1 = (f[t])^(1/2) /. Part[s, 4]

%t RealDigits[%, 10, 100] (* (A) A195286 *)

%t f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2

%t s = NSolve[D[f[t], t] == 0, t, 150]

%t f3 = (f[t])^(1/2) /. Part[s, 1]

%t RealDigits[%, 10, 100] (* (B) A195288 *)

%t f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2

%t s = NSolve[D[f[t], t] == 0, t, 150]

%t f2 = (f[t])^(1/2) /. Part[s, 4]

%t RealDigits[%, 10, 100] (* (C) A010487 *)

%t (f1 + f2 + f3)/(a + b + c)

%t RealDigits[%, 10, 100] (* Philo(A,B,C,I) A195289 *)

%Y Cf. A195284.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Sep 14 2011