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A195497 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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

%I A195497 #5 Mar 30 2012 18:57:47
%S A195497 8,6,2,9,6,8,7,9,2,1,4,1,0,3,7,4,3,4,1,3,6,0,1,0,4,3,3,0,1,6,1,7,3,1,
%T A195497 2,5,4,9,8,3,6,2,2,2,5,5,0,0,4,9,0,7,6,8,0,7,3,5,7,1,1,5,5,4,5,8,2,8,
%U A195497 9,7,8,6,0,7,8,9,7,7,8,0,1,6,6,5,7,3,0,5,7,8,9,6,9,2,3,1,2,1,2,2
%N A195497 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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).
%C A195497 See A195304 for definitions and a general discussion.
%e A195497 (C)=0.862968792141037434136010433016173...
%t A195497 a = b - 1; b = GoldenRatio; h = 2 a/3; k = b/3;
%t A195497 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195497 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195497 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195497 RealDigits[%, 10, 100] (* (A) A195495 *)
%t A195497 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195497 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195497 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195497 RealDigits[%, 10, 100] (* (B) A195496 *)
%t A195497 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195497 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195497 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195497 RealDigits[%, 10, 100] (* (C) A195497 *)
%t A195497 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195497 RealDigits[%, 10, 100] (* Philo(ABC,G) A195498 *)
%Y A195497 Cf. A195304.
%K A195497 nonn,cons
%O A195497 0,1
%A A195497 _Clark Kimberling_, Sep 19 2011