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%I #10 Dec 24 2017 09:25:55
%S 1,4,8,0,6,3,4,6,1,0,1,1,1,7,3,9,3,2,9,2,8,0,6,7,8,6,5,6,4,1,5,6,3,5,
%T 3,1,5,8,9,4,1,0,1,5,1,8,3,5,9,9,8,2,3,8,2,1,5,1,7,7,4,5,3,3,9,2,3,8,
%U 6,3,2,4,5,6,4,3,9,1,0,6,4,8,7,8,7,2,9,1,2,1,5,7,7,3,5,5,9,8,6,0
%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)=(sqrt(2),sqrt(3),sqrt(5)).
%C See A195304 for definitions and a general discussion.
%H G. C. Greubel, <a href="/A195455/b195455.txt">Table of n, a(n) for n = 1..10000</a>
%e (B)=1.48063461011173932928067865641563531589410...
%t a = Sqrt[2]; b = Sqrt[3]; 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) A195454 *)
%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) A195455 *)
%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) A195456 *)
%t c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195457 *)
%Y Cf. A195304, A195454, A195456, A195457.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Sep 19 2011
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