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A195493 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(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

%I A195493 #5 Mar 30 2012 18:57:47
%S A195493 7,5,9,3,1,0,7,7,8,3,7,3,7,3,4,9,5,6,8,1,1,8,4,2,6,9,0,4,9,7,7,6,7,3,
%T A195493 6,8,7,0,2,8,5,5,3,5,3,7,4,8,7,0,3,2,3,0,0,0,4,2,2,3,8,7,9,7,5,8,9,9,
%U A195493 1,7,4,6,7,7,7,2,2,6,0,4,6,7,1,3,9,8,3,0,8,0,4,2,3,1,3,3,2,0,1,1
%N A195493 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(r),r), where r=(1+sqrt(5))/2 (the golden ratio).
%C A195493 See A195304 for definitions and a general discussion.
%e A195493 (C)=0.759310778373734956811842690497767...
%t A195493 a = 1; b = Sqrt[GoldenRatio]; h = 2 a/3; k = b/3;
%t A195493 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195493 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195493 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195493 RealDigits[%, 10, 100] (* (A) A195491 *)
%t A195493 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195493 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195493 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195493 RealDigits[%, 10, 100] (* (B) A195492 *)
%t A195493 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195493 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195493 f3 = (f[t])^(1/2) /. Part[s, 4]
%t A195493 RealDigits[%, 10, 100] (* (C) A195493 *)
%t A195493 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195493 RealDigits[%, 10, 100] (* Philo(ABC,G) A195494 *)
%Y A195493 Cf. A195304.
%K A195493 nonn,cons
%O A195493 0,1
%A A195493 _Clark Kimberling_, Sep 19 2011