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A195412 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)=(5,12,13).

%I A195412 #5 Mar 30 2012 18:57:45
%S A195412 3,2,6,8,5,0,5,3,9,8,0,7,1,2,0,6,3,7,7,1,0,0,2,1,3,3,3,6,8,5,3,8,6,5,
%T A195412 8,9,3,3,0,4,0,5,7,8,5,6,4,9,3,4,6,0,1,3,3,6,0,9,8,0,3,4,6,4,6,2,9,0,
%U A195412 5,1,3,6,7,8,6,1,7,0,4,5,2,8,9,0,5,2,8,9,7,5,7,9,2,6,9,2,9,9,2,7
%N A195412 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)=(5,12,13).
%C A195412 See A195304 for definitions and a general discussion.
%e A195412 (A)=3.268505398071206377100213336853865893304057...
%t A195412 a = 5; b = 12; h = 2 a/3; k = b/3;
%t A195412 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195412 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195412 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195412 RealDigits[%, 10, 100] (* (A) A195412 *)
%t A195412 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195412 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195412 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195412 RealDigits[%, 10, 100] (* (B) A195413 *)
%t A195412 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195412 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195412 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195412 RealDigits[%, 10, 100] (* (C) A195414 *)
%t A195412 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195412 RealDigits[%, 10, 100] (* Philo(ABC,G) A195424 *)
%Y A195412 Cf. A195413, A195414, A195424.
%K A195412 nonn,cons
%O A195412 1,1
%A A195412 _Clark Kimberling_, Sep 18 2011