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A195435 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)=(1,2,sqrt(5)).

%I A195435 #8 Feb 11 2025 13:31:41
%S A195435 1,3,8,7,3,1,2,7,2,8,3,1,3,8,2,0,9,1,7,4,6,3,3,6,0,2,4,0,9,8,2,2,3,3,
%T A195435 2,1,2,5,9,6,4,4,1,8,6,2,5,1,6,9,7,6,7,6,7,4,7,6,5,4,1,2,8,4,2,3,5,6,
%U A195435 2,8,3,4,5,5,0,0,9,7,1,9,7,9,4,1,5,3,7,9,6,0,7,3,5,5,9,4,5,3,7,4
%N A195435 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)=(1,2,sqrt(5)).
%C A195435 See A195304 for definitions and a general discussion.
%H A195435 <a href="/index/Al#algebraic_06">Index entries for algebraic numbers, degree 6</a>.
%e A195435 (B)=1.387312728313820917463360240982233212...
%t A195435 a = 1; b = 2; h = 2 a/3; k = b/3;
%t A195435 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195435 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195435 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195435 RealDigits[%, 10, 100] (* (A) A195434 *)
%t A195435 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195435 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195435 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195435 RealDigits[%, 10, 100] (* (B) A195435 *)
%t A195435 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195435 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195435 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195435 RealDigits[%, 10, 100] (* (C) A195444 *)
%t A195435 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195435 RealDigits[%, 10, 100] (* Philo(ABC,G) A195445 *)
%o A195435 (PARI) sqrt(subst((9*t^4 - 30*t^3 + 41*t^2 - 28*t + 8)/(9*t^2 - 12*t + 4), t, polrootsreal(27*t^3 - 54*t^2 + 36*t - 4)[1])) \\ _Charles R Greathouse IV_, Feb 11 2025
%o A195435 (PARI) polrootsreal(729*x^6 - 1215*x^4 - 297*x^2 - 125)[2] \\ _Charles R Greathouse IV_, Feb 11 2025
%Y A195435 Cf. A195304.
%K A195435 nonn,cons
%O A195435 1,2
%A A195435 _Clark Kimberling_, Sep 18 2011