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%I #5 Mar 30 2012 18:57:45
%S 6,4,9,5,1,6,0,5,0,2,9,2,0,9,4,5,3,2,4,4,9,9,3,9,5,2,6,3,7,4,2,5,2,4,
%T 7,5,8,1,4,1,8,7,5,7,5,9,9,5,3,5,1,0,7,5,6,6,3,8,3,8,5,2,2,9,2,8,8,4,
%U 5,4,9,7,1,6,2,6,9,8,8,7,3,3,6,7,7,6,6,2,9,4,8,0,8,7,6,3,4,5,0,6
%N Decimal expansion of shortest length, (C), of segment from side CA through centroid to side CB in right triangle ABC with sidelengths (a,b,c)=(5,12,13).
%C See A195304 for definitions and a general discussion.
%e (C)=6.49516050292094532449939526374252475814...
%t a = 5; b = 12; 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) A195412 *)
%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) A195413 *)
%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) A195414 *)
%t c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195424 *)
%Y Cf. A195304.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Sep 18 2011
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