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%I #5 Mar 30 2012 18:57:47
%S 2,6,5,9,6,8,4,7,2,2,7,6,3,0,1,5,7,8,2,8,6,9,3,1,5,8,7,6,5,0,6,1,2,3,
%T 1,9,7,2,2,0,9,7,7,0,3,4,5,3,4,2,9,3,4,0,4,1,2,1,6,6,2,3,1,6,8,7,6,3,
%U 1,8,7,1,6,8,8,0,8,1,7,7,1,2,0,1,7,2,9,6,9,9,7,2,9,4,0,2,1,0,8,3
%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(7),3,4).
%C See A195304 for definitions and a general discussion.
%e (B)=2.659684722763015782869315876506123197220...
%t a = Sqrt[7]; b = 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) A195487 *)
%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) A195488 *)
%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) A195489 *)
%t c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195490 *)
%Y Cf. A195304.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Sep 19 2011
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