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A195301 Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(1,1,sqrt(2)).

%I A195301 #14 Jul 27 2018 09:35:51
%S A195301 6,3,4,0,5,0,6,7,1,1,2,4,4,2,8,8,5,0,6,8,5,0,5,2,8,8,5,3,4,3,9,6,2,2,
%T A195301 1,3,1,9,8,9,1,0,0,0,3,5,6,9,5,5,3,6,1,2,9,8,9,9,8,5,8,4,0,1,0,1,7,7,
%U A195301 1,7,5,8,3,2,3,6,9,1,8,9,6,9,6,3,2,4,9,4,5,6,6,6,3,1,1,0,0,0
%N A195301 Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(1,1,sqrt(2)).
%C A195301 See A195284 for definitions and a general discussion.
%H A195301 G. C. Greubel, <a href="/A195301/b195301.txt">Table of n, a(n) for n = 0..10000</a>
%e A195301 (A)=0.63405067112442885068505288534396221319891000...
%t A195301 a = 1; b = 1; c = Sqrt[2];
%t A195301 h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
%t A195301 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2;
%t A195301 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195301 f1 = (f[t])^(1/2) /. Part[s, 1]
%t A195301 RealDigits[%, 10, 100] (* (A) A195301 *)
%t A195301 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195301 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195301 f3 = (f[t])^(1/2) /. Part[s, 4]
%t A195301 RealDigits[%, 10, 100] (* (B)=(A) *)
%t A195301 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195301 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195301 f2 = (f[t])^(1/2) /. Part[s, 1]
%t A195301 RealDigits[%, 10, 100] (* (C) A163960 *)
%t A195301 (f1 + f2 + f3)/(a + b + c)
%t A195301 RealDigits[%, 10, 100]  (* Philo(ABC,I), A195303 *)
%Y A195301 Cf. A195284, A195303, A195304.
%K A195301 nonn,cons
%O A195301 0,1
%A A195301 _Clark Kimberling_, Sep 14 2011