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%I
%S 6,3,5,0,7,4,3,6,8,6,2,0,6,6,8,1,3,7,5,6,2,1,5,7,6,6,1,6,4,5,4,6,4,6,
%T 0,8,6,9,7,6,8,0,5,0,0,0,7,5,5,5,1,9,3,1,3,2,1,8,6,7,4,2,2,9,2,7,5,7,
%U 4,9,4,0,4,3,3,5,5,5,9,7,7,8,3,2,0,1,1,3,4,1,5,5,5,7,0,6,3,9,7,8
%N Decimal expansion of shortest length, (A), of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(2),sqrt(3)).
%C See A195304 for definitions and a general discussion.
%e (A)=0.6350743686206681375621576616454646086976805000...
%t a = 1; b = Sqrt[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) A195471 *)
%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) A195472 *)
%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) A195473 *)
%t c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195474 *)
%Y Cf. A195304, A195471, A195472, A195473.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Sep 19 2011
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