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A195447 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)=(1,3,sqrt(10)).

%I A195447 #5 Mar 30 2012 18:57:45
%S A195447 1,8,0,1,8,6,6,2,4,2,4,7,4,4,9,4,6,0,8,4,7,6,8,6,9,5,5,0,8,9,8,5,4,1,
%T A195447 3,2,7,3,2,0,1,3,0,8,1,3,4,9,9,3,3,7,6,7,8,8,7,9,9,2,0,1,0,7,7,9,9,5,
%U A195447 3,6,2,0,2,8,4,4,8,8,8,3,8,7,8,2,5,1,0,5,9,0,6,4,2,0,4,3,1,0,2,9
%N A195447 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)=(1,3,sqrt(10)).
%C A195447 See A195304 for definitions and a general discussion.
%e A195447 (B)=1.8018662424744946084768695508...
%t A195447 a = 1; b = 3; h = 2 a/3; k = b/3;
%t A195447 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195447 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195447 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195447 RealDigits[%, 10, 100] (* (A) A195446  *)
%t A195447 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195447 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195447 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195447 RealDigits[%, 10, 100] (* (B) A195447 *)
%t A195447 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195447 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195447 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195447 RealDigits[%, 10, 100] (* (C) A195448 *)
%t A195447 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195447 RealDigits[%, 10, 100] (* Philo(ABC,G) A195449 *)
%Y A195447 Cf. A195304.
%K A195447 nonn,cons
%O A195447 1,2
%A A195447 _Clark Kimberling_, Sep 18 2011