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A195303 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 1,1,sqrt(2) right triangle ABC.

%I A195303 #16 Jul 27 2018 09:35:59
%S A195303 6,1,4,0,5,8,9,7,1,0,3,2,2,1,2,6,1,1,5,4,6,3,8,4,8,9,2,5,3,9,3,8,5,4,
%T A195303 0,8,2,6,0,3,6,7,3,8,6,8,9,6,9,9,6,8,9,2,7,6,4,7,9,4,1,9,1,7,6,7,3,2,
%U A195303 8,5,7,4,5,1,7,0,3,8,0,3,8,4,9,2,8,5,5,8,3,1,6,0,3,1,2,0,5,5,1,2
%N A195303 Decimal expansion of normalized Philo sum, Philo(ABC,I), where I=incenter of a 1,1,sqrt(2) right triangle ABC.
%C A195303 See A195284 for definitions and a general discussion.  This constant is the maximum of Philo(ABC,I) over all triangles ABC.
%H A195303 G. C. Greubel, <a href="/A195303/b195303.txt">Table of n, a(n) for n = 0..10000</a>
%F A195303 Equals (3*sqrt(2)-4)*(1+2*sqrt(2-sqrt(2))).
%e A195303 Philo(ABC,I)=0.614058971032212611546384892539385408260...
%t A195303 a = 1; b = 1; c = Sqrt[2];
%t A195303 h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
%t A195303 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2;
%t A195303 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195303 f1 = (f[t])^(1/2) /. Part[s, 1]
%t A195303 RealDigits[%, 10, 100] (* (A) A195301 *)
%t A195303 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195303 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195303 f3 = (f[t])^(1/2) /. Part[s, 4]
%t A195303 RealDigits[%, 10, 100] (* (B)=(A) *)
%t A195303 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195303 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195303 f2 = (f[t])^(1/2) /. Part[s, 1]
%t A195303 RealDigits[%, 10, 100] (* (C) A163960 *)
%t A195303 (f1 + f2 + f3)/(a + b + c)
%t A195303 RealDigits[%, 10, 100]  (* Philo(ABC,I), A195303 *)
%o A195303 (PARI) (3*sqrt(2)-4)*(1+2*sqrt(2-sqrt(2))) \\ _Michel Marcus_, Jul 27 2018
%Y A195303 Cf. A195284.
%K A195303 nonn,cons
%O A195303 0,1
%A A195303 _Clark Kimberling_, Sep 14 2011