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A195292 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 7,24,25 right triangle ABC.

%I A195292 #10 Mar 30 2012 18:57:45
%S A195292 3,9,3,6,8,2,0,8,2,8,8,4,8,5,4,1,9,2,6,3,7,0,4,4,8,6,7,7,1,1,9,8,5,3,
%T A195292 6,1,3,6,9,9,3,9,7,3,2,2,1,2,0,9,2,5,0,3,2,3,6,5,3,3,0,1,3,1,0,0,3,3,
%U A195292 8,6,1,8,4,9,3,0,4,0,0,6,8,3,6,0,2,7,5,2,6,1,4,0,7,1,1,7,8,3,9,8
%N A195292 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 7,24,25 right triangle ABC.
%C A195292 See A195284 for definitions and a general discussion.
%e A195292 Philo(ABC,I)=0.39368208288485419263704486771198536...
%t A195292 a = 7; b = 24; c = 25;
%t A195292 h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
%t A195292 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2;
%t A195292 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195292 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195292 RealDigits[%, 10, 100] (* (A) A195290 *)
%t A195292 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195292 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195292 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195292 RealDigits[%, 10, 100] (* (B)=7.5 *)
%t A195292 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195292 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195292 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195292 RealDigits[%, 10, 100] (* (C) A010524 *)
%t A195292 (f1 + f2 + f3)/(a + b + c)
%t A195292 RealDigits[%, 10, 100] (* Philo(ABC,I) A195292 *)
%Y A195292 Cf. A195284.
%K A195292 nonn,cons
%O A195292 0,1
%A A195292 _Clark Kimberling_, Sep 14 2011