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A195496 Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

%I A195496 #5 Mar 30 2012 18:57:47
%S A195496 1,0,1,7,1,5,3,4,4,6,7,5,4,8,0,4,4,6,6,2,5,6,7,9,8,1,8,7,8,1,6,6,0,6,
%T A195496 3,3,6,9,7,4,3,6,7,9,8,2,5,5,3,7,4,6,3,9,5,6,4,0,3,4,9,5,5,6,1,7,5,7,
%U A195496 7,6,1,4,7,5,2,9,8,5,3,2,8,9,2,4,2,4,6,6,6,3,7,8,4,1,8,4,8,3,0,3
%N A195496 Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).
%C A195496 See A195304 for definitions and a general discussion.
%e A195496 (B)=1.017153446754804466256798187816606336...
%t A195496 a = b - 1; b = GoldenRatio; h = 2 a/3; k = b/3;
%t A195496 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195496 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195496 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195496 RealDigits[%, 10, 100] (* (A) A195495 *)
%t A195496 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195496 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195496 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195496 RealDigits[%, 10, 100] (* (B) A195496 *)
%t A195496 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195496 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195496 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195496 RealDigits[%, 10, 100] (* (C) A195497 *)
%t A195496 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195496 RealDigits[%, 10, 100] (* Philo(ABC,G) A195498 *)
%Y A195496 Cf. A195304.
%K A195496 nonn,cons
%O A195496 1,4
%A A195496 _Clark Kimberling_, Sep 19 2011