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%I #6 Mar 30 2012 18:57:45
%S 1,1,1,5,7,6,4,8,6,9,4,3,5,2,7,6,7,1,8,6,7,5,9,0,8,9,5,5,5,1,8,7,7,7,
%T 3,5,1,2,3,8,9,1,0,5,1,4,0,2,3,5,9,5,2,4,1,1,9,9,6,4,0,3,7,9,6,8,4,1,
%U 2,6,3,7,5,2,0,3,6,5,0,4,8,6,0,7,9,5,8,1,1,3,9,7,2,3,4,1,9,6,9,1
%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)=(1,2,sqrt(5)).
%C See A195304 for definitions and a general discussion.
%e (C)=1.11576486943527671867590895551877735123891...
%t a = 1; b = 2; 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) A195434 *)
%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) A195435 *)
%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) A195444 *)
%t c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195445 *)
%Y Cf. A195304.
%K nonn,cons
%O 1,4
%A _Clark Kimberling_, Sep 18 2011