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%I #8 Jan 27 2018 02:29:31
%S 9,0,5,3,4,7,0,9,3,0,8,3,6,4,7,2,1,7,2,3,6,0,7,6,5,7,6,7,8,5,6,8,4,5,
%T 4,6,1,7,8,0,0,6,3,3,9,6,0,4,8,0,3,3,7,3,8,2,0,9,5,3,7,3,3,6,5,1,5,7,
%U 8,5,9,6,6,5,7,7,8,9,2,5,8,5,0,0,9,0,3,9,2,4,7,4,0,7,0,6,2,6,8,1
%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)=(sqrt(2),sqrt(5),sqrt(7)).
%C See A195304 for definitions and a general discussion.
%H G. C. Greubel, <a href="/A195483/b195483.txt">Table of n, a(n) for n = 0..10000</a>
%e (A)=0.90534709308364721723607657678568...
%t a = Sqrt[2]; b = Sqrt[5]; 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) A195483 *)
%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) A195484 *)
%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) A195485 *)
%t c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195486 *)
%Y Cf. A195304, A195484, A195485, A195486.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Sep 19 2011
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