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A195286 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)=(5,12,13).

%I A195286 #10 Mar 30 2012 18:57:45
%S A195286 4,0,7,9,2,1,5,6,1,0,8,7,4,2,2,7,8,6,4,0,2,2,5,7,9,2,8,7,2,1,8,2,2,5,
%T A195286 5,9,1,6,5,1,0,1,6,7,5,6,8,7,9,6,7,7,1,2,6,0,6,7,9,7,6,6,4,3,5,4,0,7,
%U A195286 4,6,9,0,5,6,4,9,7,7,9,3,5,6,4,7,0,6,7,9,0,8,0,8,7,5,1,4,5,6,0,6
%N A195286 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)=(5,12,13).
%C A195286 See A195284 for definitions and a general discussion.
%e A195286 (A)=4.0792156108742278640225792872182255...
%t A195286 a = 5; b = 12; c = 13;
%t A195286 h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
%t A195286 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2;
%t A195286 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195286 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195286 RealDigits[%, 10, 100] (* (A) A195286 *)
%t A195286 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195286 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195286 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195286 RealDigits[%, 10, 100] (* (B) A195288 *)
%t A195286 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195286 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195286 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195286 RealDigits[%, 10, 100] (* (C) A010487 *)
%t A195286 (f1 + f2 + f3)/(a + b + c)
%t A195286 RealDigits[%, 10, 100] (* Philo(A,B,C,I) A195289 *)
%Y A195286 Cf. A195284, A195288, A195289.
%K A195286 nonn,cons
%O A195286 1,1
%A A195286 _Clark Kimberling_, Sep 14 2011