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A195475 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)=(1,sqrt(3),2) and angles 30,60,90.

%I A195475 #5 Mar 30 2012 18:57:45
%S A195475 6,4,3,8,4,6,3,1,3,2,9,8,7,4,3,5,3,1,5,6,9,3,7,2,1,0,7,2,1,1,8,0,9,7,
%T A195475 2,0,6,7,5,1,9,8,1,6,0,8,2,1,8,5,8,7,2,8,7,9,9,8,8,4,7,9,2,4,7,7,6,0,
%U A195475 4,9,3,3,7,6,7,7,9,9,8,3,9,1,9,0,0,8,7,9,2,8,3,1,3,7,8,0,4,6,5,7
%N A195475 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)=(1,sqrt(3),2) and angles 30,60,90.
%C A195475 See A195304 for definitions and a general discussion.
%e A195475 (A)=0.643846313298743531569372107211809720...
%t A195475 a = 1; b = Sqrt[3]; h = 2 a/3; k = b/3;
%t A195475 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195475 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195475 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195475 RealDigits[%, 10, 100] (* (A) A195575 *)
%t A195475 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195475 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195475 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195475 RealDigits[%, 10, 100] (* (B) A195576 *)
%t A195475 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195475 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195475 f3 = (f[t])^(1/2) /. Part[s, 4]
%t A195475 RealDigits[%, 10, 100] (* (C) A195577 *)
%t A195475 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195475 RealDigits[%, 10, 100] (* Philo(ABC,G) A195578 *)
%Y A195475 Cf. A195304, A195476, A195477, A195478.
%K A195475 nonn,cons
%O A195475 0,1
%A A195475 _Clark Kimberling_, Sep 19 2011