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A195306 Decimal expansion of shortest length, (C), of segment from side CA through centroid to side CB in right triangle ABC with sidelengths (a,b,c)=(3,4,5).

%I A195306 #5 Mar 30 2012 18:57:45
%S A195306 2,3,7,2,5,9,1,6,7,4,9,5,6,7,4,9,3,0,8,0,7,5,0,9,8,5,2,9,9,4,0,3,2,0,
%T A195306 1,5,0,0,5,7,3,6,1,3,2,7,0,9,0,4,6,2,3,9,2,7,4,9,3,7,2,8,5,0,0,3,7,0,
%U A195306 0,1,9,2,5,1,5,1,9,5,1,6,9,6,7,1,2,1,8,5,7,5,2,0,3,1,2,4,7,1,2,4
%N A195306 Decimal expansion of shortest length, (C), of segment from side CA through centroid to side CB in right triangle ABC with sidelengths (a,b,c)=(3,4,5).
%C A195306 See A195304 for definitions and a general discussion.
%e A195306 (C)=2.3725916749567493080750985299403201500573613270...
%t A195306 a = 3; b = 4; h = 2 a/3; k = b/3;
%t A195306 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195306 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195306 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195306 RealDigits[%, 10, 100]   (* (A) A195304 *)
%t A195306 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195306 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195306 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195306 RealDigits[%, 10, 100]   (* (B) A195305 *)
%t A195306 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195306 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195306 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195306 RealDigits[%, 10, 100]   (* (C) A195306 *)
%t A195306 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195306 RealDigits[%, 10, 100]   (* Philo(ABC,G) A195411 *)
%Y A195306 Cf. A195304.
%K A195306 nonn,cons
%O A195306 1,1
%A A195306 _Clark Kimberling_, Sep 18 2011