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A195451 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)=(2,3,sqrt(13)).

%I A195451 #5 Mar 30 2012 18:57:45
%S A195451 2,3,4,1,1,6,0,7,9,3,0,7,3,3,2,1,9,7,5,6,1,4,8,5,3,3,8,1,3,7,6,3,8,3,
%T A195451 4,9,3,4,2,4,4,2,5,3,8,8,6,7,8,4,4,6,8,7,5,5,7,4,6,5,0,5,2,2,5,5,1,8,
%U A195451 6,1,7,6,4,9,3,5,2,2,3,4,7,9,6,5,6,5,8,7,1,7,9,4,6,1,0,5,3,0,3,9
%N A195451 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)=(2,3,sqrt(13)).
%C A195451 See A195304 for definitions and a general discussion.
%e A195451 (B)=2.3411607930733219756148533813763834934244253886784...
%t A195451 a = 2; b = 3; h = 2 a/3; k = b/3;
%t A195451 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195451 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195451 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195451 RealDigits[%, 10, 100] (* (A) A195450 *)
%t A195451 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195451 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195451 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195451 RealDigits[%, 10, 100] (* (B) A195451 *)
%t A195451 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195451 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195451 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195451 RealDigits[%, 10, 100] (* (C) A195452  *)
%t A195451 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195451 RealDigits[%, 10, 100] (* Philo(ABC,G) A195453 *)
%Y A195451 Cf. A195304.
%K A195451 nonn,cons
%O A195451 1,1
%A A195451 _Clark Kimberling_, Sep 18 2011