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A195472 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)=(1,sqrt(2),sqrt(3)).

%I A195472 #8 Jan 27 2018 02:29:23
%S A195472 1,1,3,2,4,4,8,9,8,3,6,7,2,5,6,4,4,8,0,4,2,5,9,7,1,2,5,1,8,3,3,8,0,3,
%T A195472 5,9,6,8,2,9,8,2,7,8,2,9,1,7,5,7,2,5,8,7,9,4,6,3,3,8,7,3,8,2,7,8,3,1,
%U A195472 4,6,7,6,3,1,5,0,5,5,9,5,0,5,5,3,6,6,3,7,1,0,8,6,8,9,0,6,1,5,3,1
%N A195472 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)=(1,sqrt(2),sqrt(3)).
%C A195472 See A195304 for definitions and a general discussion.
%H A195472 G. C. Greubel, <a href="/A195472/b195472.txt">Table of n, a(n) for n = 1..10000</a>
%e A195472 (B)=1.1324489836725644804259712518338035968298278...
%t A195472 a = 1; b = Sqrt[2]; h = 2 a/3; k = b/3;
%t A195472 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195472 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195472 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195472 RealDigits[%, 10, 100] (* (A) A195471 *)
%t A195472 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195472 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195472 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195472 RealDigits[%, 10, 100] (* (B) A195472 *)
%t A195472 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195472 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195472 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195472 RealDigits[%, 10, 100] (* (C) A195473 *)
%t A195472 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195472 RealDigits[%, 10, 100] (* Philo(ABC,G) A195474 *)
%Y A195472 Cf. A195304.
%K A195472 nonn,cons
%O A195472 1,3
%A A195472 _Clark Kimberling_, Sep 19 2011