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A195288 Decimal expansion of shortest length, (C), of segment from side CA through incenter to side CB in right triangle ABC with sidelengths (a,b,c)=(5,12,13).

%I A195288 #10 Mar 30 2012 18:57:45
%S A195288 4,8,0,7,4,0,1,7,0,0,6,1,8,6,5,2,3,9,0,8,2,5,6,2,8,3,5,6,6,2,7,3,2,7,
%T A195288 9,2,8,3,3,5,0,6,2,0,9,8,4,6,0,3,2,8,2,8,3,6,1,3,9,3,7,4,0,8,3,0,2,8,
%U A195288 8,9,2,6,4,3,9,0,6,8,0,5,9,3,6,0,6,1,5,8,7,7,6,0,2,4,6,5,4,2,9,0
%N A195288 Decimal expansion of shortest length, (C), of segment from side CA through incenter to side CB in right triangle ABC with sidelengths (a,b,c)=(5,12,13).
%C A195288 See A195284 for definitions and a general discussion.
%e A195288 (C)=4.80740170061865239082562835...
%t A195288 a = 5; b = 12; c = 13;
%t A195288 h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
%t A195288 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2;
%t A195288 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195288 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195288 RealDigits[%, 10, 100] (* (A) A195286 *)
%t A195288 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195288 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195288 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195288 RealDigits[%, 10, 100] (* (B) A195288 *)
%t A195288 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195288 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195288 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195288 RealDigits[%, 10, 100] (* (C) A010487 *)
%t A195288 (f1 + f2 + f3)/(a + b + c)
%t A195288 RealDigits[%, 10, 100] (* Philo(A,B,C,I) A195289 *)
%Y A195288 Cf. A195284.
%K A195288 nonn,cons
%O A195288 1,1
%A A195288 _Clark Kimberling_, Sep 14 2011