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A195434 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)=(1,2,sqrt(5)).

%I A195434 #7 Feb 11 2025 13:15:58
%S A195434 6,4,8,7,8,2,1,3,4,1,2,6,1,6,1,2,8,5,3,8,8,8,0,3,0,3,8,0,7,6,6,9,3,5,
%T A195434 6,0,6,1,9,4,0,3,5,5,7,0,5,8,6,7,9,5,2,3,3,9,6,4,1,2,8,3,6,3,6,8,3,3,
%U A195434 2,9,8,5,3,3,9,6,2,2,6,7,3,0,3,5,9,1,4,7,7,3,5,6,8,8,4,0,8,0,4,7
%N A195434 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)=(1,2,sqrt(5)).
%C A195434 See A195304 for definitions and a general discussion.
%H A195434 <a href="/index/Al#algebraic_06">Index entries for algebraic numbers, degree 6</a>.
%e A195434 (A)=0.648782134126161285388803038076693560619403...
%t A195434 a = 1; b = 2; h = 2 a/3; k = b/3;
%t A195434 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195434 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195434 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195434 RealDigits[%, 10, 100] (* (A) A195434 *)
%t A195434 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195434 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195434 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195434 RealDigits[%, 10, 100] (* (B) A195435 *)
%t A195434 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195434 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195434 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195434 RealDigits[%, 10, 100] (* (C) A195444 *)
%t A195434 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195434 RealDigits[%, 10, 100] (* Philo(ABC,G) A195445 *)
%o A195434 (PARI) sqrt(subst((45*t^4 - 126*t^3 + 125*t^2 - 52*t + 8)/(9*t^2 - 12*t + 4), t, polrootsreal(270*x^4 - 738*x^3 + 756*x^2 - 344*x + 56)[1])) \\ _Charles R Greathouse IV_, Feb 11 2025
%o A195434 (PARI) polrootsreal(3645*x^6 + 11421*x^4 + 6219*x^2 - 4913)[2] \\ _Charles R Greathouse IV_, Feb 11 2025
%Y A195434 Cf. A195304, A195435, A195444, A195445.
%K A195434 nonn,cons
%O A195434 0,1
%A A195434 _Clark Kimberling_, Sep 18 2011