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A195495 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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

%I A195495 #5 Mar 30 2012 18:57:47
%S A195495 4,0,5,1,7,7,8,2,9,7,2,0,5,7,1,7,7,7,8,8,0,3,0,7,3,9,4,9,8,0,5,1,4,5,
%T A195495 8,4,6,8,8,3,2,3,9,3,7,4,0,8,9,2,3,7,6,9,9,0,7,8,5,6,5,8,0,7,3,8,9,5,
%U A195495 8,9,0,4,6,6,4,6,2,1,3,2,6,2,2,8,4,4,7,6,8,9,3,7,6,0,2,9,7,1,8,5
%N A195495 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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).
%C A195495 See A195304 for definitions and a general discussion.
%e A195495 (A)=0.40517782972057177788030739498051458468832393740...
%t A195495 a = b - 1; b = GoldenRatio; h = 2 a/3; k = b/3;
%t A195495 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195495 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195495 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195495 RealDigits[%, 10, 100] (* (A) A195495 *)
%t A195495 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195495 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195495 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195495 RealDigits[%, 10, 100] (* (B) A195496 *)
%t A195495 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195495 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195495 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195495 RealDigits[%, 10, 100] (* (C) A195497 *)
%t A195495 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195495 RealDigits[%, 10, 100] (* Philo(ABC,G) A195498 *)
%Y A195495 Cf. A195304, A195496, A195497, A195498.
%K A195495 nonn,cons
%O A195495 0,1
%A A195495 _Clark Kimberling_, Sep 19 2011