cp's OEIS Frontend

A195491 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(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

%I A195491 #5 Mar 30 2012 18:57:47
%S A195491 6,2,9,5,8,1,0,6,1,3,8,7,7,1,6,0,4,4,0,4,5,4,9,5,8,7,5,6,8,8,5,4,0,6,
%T A195491 9,2,2,3,1,6,8,4,9,0,8,3,8,6,6,0,7,0,2,9,6,5,1,1,2,3,1,3,4,9,6,2,5,2,
%U A195491 6,6,6,5,0,5,1,3,5,9,2,3,4,6,8,8,9,9,4,9,2,9,6,9,8,9,0,2,8,7,6,7
%N A195491 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(r),r), where r=(1+sqrt(5))/2 (the golden ratio).
%C A195491 See A195304 for definitions and a general discussion.
%e A195491 (A)=0.62958106138771604404549587568854069...
%t A195491 a = 1; b = Sqrt[GoldenRatio]; h = 2 a/3; k = b/3;
%t A195491 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195491 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195491 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195491 RealDigits[%, 10, 100] (* (A) A195491 *)
%t A195491 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195491 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195491 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195491 RealDigits[%, 10, 100] (* (B) A195492 *)
%t A195491 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195491 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195491 f3 = (f[t])^(1/2) /. Part[s, 4]
%t A195491 RealDigits[%, 10, 100] (* (C) A195493 *)
%t A195491 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195491 RealDigits[%, 10, 100] (* Philo(ABC,G) A195494 *)
%Y A195491 Cf. A195304, A195492, A195493, A195494.
%K A195491 nonn,cons
%O A195491 0,1
%A A195491 _Clark Kimberling_, Sep 19 2011