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A195296 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)=(8,15,17).

%I A195296 #10 Mar 30 2012 18:57:45
%S A195296 6,9,9,7,1,4,2,2,7,3,8,1,4,3,6,0,5,6,5,0,4,8,9,8,3,4,5,3,0,5,4,6,9,9,
%T A195296 6,9,1,8,2,5,6,7,8,0,0,1,8,6,3,1,6,6,3,4,5,3,4,0,0,0,8,0,9,3,8,4,1,3,
%U A195296 7,2,0,7,4,3,9,6,0,5,5,1,5,3,1,9,8,2,8,8,3,9,1,7,4,3,6,4,2,4,7
%N A195296 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)=(8,15,17).
%C A195296 See A195284 for definitions and a general discussion.
%e A195296 (C)=6.99714227381436056504898345305469969182567800...
%t A195296 a = 8; b = 15; c = 17;
%t A195296 h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
%t A195296 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2;
%t A195296 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195296 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195296 RealDigits[%, 10, 100] (* (A) A195293 *)
%t A195296 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195296 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195296 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195296 RealDigits[%, 10, 100] (* (B) A195296 *)
%t A195296 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195296 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195296 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195296 RealDigits[%, 10, 100] (* (C) A010524 *)
%t A195296 (f1 + f2 + f3)/(a + b + c)
%t A195296 RealDigits[%, 10, 100] (* Philo(ABC,I), A195297 *)
%Y A195296 Cf. A195284.
%K A195296 nonn,cons
%O A195296 1,1
%A A195296 _Clark Kimberling_, Sep 14 2011