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A195297 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of an 8,15,17 right triangle ABC.

%I A195297 #10 Mar 30 2012 18:57:45
%S A195297 5,4,1,6,7,7,0,5,2,1,6,1,9,8,5,5,4,2,0,6,4,7,8,0,7,6,4,5,5,6,8,5,0,0,
%T A195297 9,2,5,2,4,1,1,2,7,0,2,3,0,4,6,3,2,1,3,5,8,9,9,9,5,0,9,2,2,0,3,5,7,0,
%U A195297 4,9,6,1,6,1,6,8,7,8,2,4,4,4,1,7,0,6,0,2,2,6,8,4,8,1,3,7,9,5,8,9
%N A195297 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of an 8,15,17 right triangle ABC.
%C A195297 See A195284 for definitions and a general discussion.
%e A195297 Philo(ABC,I)=0.54167705216198554206478076455685009252411270...
%t A195297 a = 8; b = 15; c = 17;
%t A195297 h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
%t A195297 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2;
%t A195297 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195297 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195297 RealDigits[%, 10, 100] (* (A) A195293 *)
%t A195297 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195297 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195297 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195297 RealDigits[%, 10, 100] (* (B) A195296 *)
%t A195297 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195297 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195297 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195297 RealDigits[%, 10, 100] (* (C) A010524 *)
%t A195297 (f1 + f2 + f3)/(a + b + c)
%t A195297 RealDigits[%, 10, 100] (* Philo(ABC,I), A195297 *)
%Y A195297 Cf. A195284.
%K A195297 nonn,cons
%O A195297 0,1
%A A195297 _Clark Kimberling_, Sep 14 2011