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