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A195492 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)=(1,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

%I A195492 #7 Nov 26 2024 21:00:16
%S A195492 1,0,6,8,4,7,3,0,1,7,1,9,8,6,1,0,0,3,5,7,5,3,9,8,2,0,2,5,9,9,5,6,9,6,
%T A195492 7,6,0,9,5,0,5,7,8,8,0,8,5,2,8,7,9,5,2,7,3,0,6,8,4,1,0,7,5,1,9,4,3,6,
%U A195492 2,1,9,6,3,2,3,0,8,7,8,6,8,7,7,9,0,6,8,1,0,2,4,5,0,6,6,9,5,6,1,4
%N A195492 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)=(1,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).
%C A195492 See A195304 for definitions and a general discussion.
%H A195492 <a href="/index/Al#algebraic_08">Index entries for algebraic numbers, degree 8</a>
%e A195492 (B)=1.0684730171986100357539820259956...
%t A195492 a = 1; b = Sqrt[GoldenRatio]; h = 2 a/3; k = b/3;
%t A195492 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195492 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195492 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195492 RealDigits[%, 10, 100] (* (A) A195491 *)
%t A195492 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195492 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195492 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195492 RealDigits[%, 10, 100] (* (B) A195492 *)
%t A195492 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195492 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195492 f3 = (f[t])^(1/2) /. Part[s, 4]
%t A195492 RealDigits[%, 10, 100] (* (C) A195493 *)
%t A195492 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195492 RealDigits[%, 10, 100] (* Philo(ABC,G) A195494 *)
%o A195492 (PARI) polrootsreal(3464*x^8 - 16028*x^7 + 40544*x^6 + 47079*x^5 - 64883*x^4 + 50115*x^3 - 43008*x^2 - 60658*x + 32292)[4] \\ _Charles R Greathouse IV_, Nov 26 2024
%Y A195492 Cf. A195304.
%K A195492 nonn,cons
%O A195492 1,3
%A A195492 _Clark Kimberling_, Sep 19 2011