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A195477 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)=(1,sqrt(3),2).

%I A195477 #5 Mar 30 2012 18:57:47
%S A195477 9,8,8,6,5,9,2,6,2,9,8,1,9,3,8,8,4,1,7,1,3,0,9,5,8,6,3,8,8,3,8,2,5,2,
%T A195477 4,0,3,0,6,3,3,4,0,6,3,5,4,4,3,7,8,5,1,8,2,0,8,1,0,0,4,8,2,6,1,1,8,6,
%U A195477 8,8,8,2,0,4,0,6,8,1,2,5,5,6,8,6,4,5,6,7,3,2,1,8,6,2,9,0,6,8,2,4
%N A195477 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)=(1,sqrt(3),2).
%C A195477 See A195304 for definitions and a general discussion.
%e A195477 (C)=0.98865926298193884171309586388382524030...
%t A195477 a = 1; b = Sqrt[3]; h = 2 a/3; k = b/3;
%t A195477 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195477 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195477 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195477 RealDigits[%, 10, 100] (* (A) A195575 *)
%t A195477 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195477 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195477 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195477 RealDigits[%, 10, 100] (* (B) A195576 *)
%t A195477 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195477 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195477 f3 = (f[t])^(1/2) /. Part[s, 4]
%t A195477 RealDigits[%, 10, 100] (* (C) A195577 *)
%t A195477 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195477 RealDigits[%, 10, 100] (* Philo(ABC,G) A195578 *)
%Y A195477 Cf. A195304.
%K A195477 nonn,cons
%O A195477 0,1
%A A195477 _Clark Kimberling_, Sep 19 2011