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A195454 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(3),sqrt(5)).

%I A195454 #11 Dec 24 2017 09:24:18
%S A195454 8,8,7,3,6,6,3,8,9,6,4,8,5,9,1,6,1,8,6,2,7,9,8,1,8,0,5,9,7,3,8,0,8,7,
%T A195454 5,8,1,3,5,9,3,9,8,5,0,2,4,3,8,6,0,9,1,1,2,1,6,9,3,1,1,7,8,6,5,9,7,8,
%U A195454 4,4,4,6,5,3,6,2,5,9,2,1,0,0,7,0,0,8,7,0,0,3,9,6,5,8,9,1,5,1,1,2
%N A195454 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(3),sqrt(5)).
%C A195454 See A195304 for definitions and a general discussion.
%H A195454 G. C. Greubel, <a href="/A195454/b195454.txt">Table of n, a(n) for n = 0..10000</a>
%e A195454 (A)=0.88736638964859161862798180597380875813593985...
%t A195454 a = Sqrt[2]; b = Sqrt[3]; h = 2 a/3; k = b/3;
%t A195454 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195454 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195454 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195454 RealDigits[%, 10, 100] (* (A) A195454 *)
%t A195454 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195454 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195454 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195454 RealDigits[%, 10, 100] (* (B) A195455 *)
%t A195454 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195454 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195454 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195454 RealDigits[%, 10, 100] (* (C) A195456 *)
%t A195454 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195454 RealDigits[%, 10, 100] (* Philo(ABC,G) A195457 *)
%Y A195454 Cf. A195304, A195455, A195456, A195457.
%K A195454 nonn,cons
%O A195454 0,1
%A A195454 _Clark Kimberling_, Sep 19 2011