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%I #5 Mar 30 2012 18:57:45
%S 3,2,8,8,5,5,4,1,8,5,1,4,5,0,3,0,0,6,4,1,8,2,8,4,8,1,0,8,8,9,6,3,5,1,
%T 4,1,4,3,6,1,5,8,3,8,2,3,0,3,0,2,0,1,0,6,8,3,5,6,7,4,9,8,8,8,1,7,1,4,
%U 7,4,0,4,6,4,1,6,1,2,7,9,2,6,9,2,1,8,7,6,8,0,7,2,8,8,8,3,4,5,4,0
%N Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(3,4,5).
%C See A195304 for definitions and a general discussion.
%e (B)=3.288554185145030064182848108896351414361583823030...
%t a = 3; b = 4; h = 2 a/3; k = b/3;
%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) A195304 *)
%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] (* (B) A195305 *)
%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] (* (C) A195306 *)
%t c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195411 *)
%Y Cf. A195304.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Sep 18 2011
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