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A195300 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 28,45,53 right triangle ABC.

%I A195300 #8 Mar 30 2012 18:57:45
%S A195300 5,7,1,1,4,0,9,7,8,6,3,4,2,6,2,1,6,8,6,1,9,2,0,8,1,0,8,5,8,7,3,9,5,1,
%T A195300 2,2,0,5,7,8,9,8,6,2,7,5,0,4,1,0,6,1,3,2,5,5,5,4,2,1,2,6,9,6,0,1,3,0,
%U A195300 9,2,0,3,0,4,0,3,6,6,3,6,8,1,4,2,4,7,2,9,1,6,1,9,5,1,9,4,5,5,5
%N A195300 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 28,45,53 right triangle ABC.
%C A195300 See A195284 for definitions and a general discussion.
%e A195300 Philo(ABC,I)=0.5711409786342621686192081085873951220...
%t A195300 a = 28; b = 45; c = 53;
%t A195300 h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
%t A195300 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2;
%t A195300 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195300 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195300 RealDigits[%, 10, 100] (* (A) A195298 *)
%t A195300 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195300 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195300 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195300 RealDigits[%, 10, 100] (* (B) A195299 *)
%t A195300 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195300 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195300 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195300 RealDigits[%, 10, 100] (* (C)=20*sqrt(2) *)
%t A195300 (f1 + f2 + f3)/(a + b + c)
%t A195300 RealDigits[%, 10, 100]  (* Phil(ABC,I), A195300 *)
%Y A195300 Cf. A195284.
%K A195300 nonn,cons
%O A195300 0,1
%A A195300 _Clark Kimberling_, Sep 14 2011