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A195450 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)=(2,3,sqrt(13)).

%I A195450 #5 Mar 30 2012 18:57:45
%S A195450 1,2,7,5,7,0,6,9,9,4,4,4,0,0,5,5,2,7,6,4,5,0,3,7,8,5,5,6,2,9,1,5,3,5,
%T A195450 2,8,7,5,2,2,8,4,4,7,8,4,4,9,8,3,3,3,9,8,7,9,3,6,7,0,3,0,2,3,1,4,9,2,
%U A195450 5,0,0,7,8,6,0,5,6,3,7,3,4,3,6,0,6,4,1,4,5,3,9,6,2,7,5,9,0,9,2,4
%N A195450 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)=(2,3,sqrt(13)).
%C A195450 See A195304 for definitions and a general discussion.
%e A195450 (A)=1.2757069944400552764503785562915352875228447844...
%t A195450 a = 2; b = 3; h = 2 a/3; k = b/3;
%t A195450 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195450 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195450 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195450 RealDigits[%, 10, 100] (* (A) A195450 *)
%t A195450 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195450 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195450 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195450 RealDigits[%, 10, 100] (* (B) A195451 *)
%t A195450 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195450 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195450 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195450 RealDigits[%, 10, 100] (* (C) A195452  *)
%t A195450 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195450 RealDigits[%, 10, 100] (* Philo(ABC,G) A195453 *)
%Y A195450 Cf. A195304, A195451, A195452, A195453.
%K A195450 nonn,cons
%O A195450 1,2
%A A195450 _Clark Kimberling_, Sep 18 2011