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A195433 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)=(1,1,sqrt(2)).

%I A195433 #9 Jul 27 2018 09:36:08
%S A195433 6,1,4,7,5,7,2,2,7,2,3,3,3,9,0,6,2,1,5,9,3,3,1,9,2,4,8,0,9,1,1,9,0,0,
%T A195433 9,9,4,7,1,1,6,2,5,4,4,6,2,5,6,9,8,3,6,3,8,5,8,2,6,4,6,7,2,6,2,1,6,2,
%U A195433 5,6,1,1,4,6,1,7,9,6,2,0,4,1,6,1,6,8,8,1,5,6,9,9,9,1,9,3,9,5,0,1
%N A195433 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)=(1,1,sqrt(2)).
%C A195433 See A195304 for definitions and a general discussion.
%H A195433 G. C. Greubel, <a href="/A195433/b195433.txt">Table of n, a(n) for n = 0..10000</a>
%e A195433 (A)=0.6147572272333906215933192480911...
%t A195433 a = 1; b = 1; h = 2 a/3; k = b/3;
%t A195433 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195433 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195433 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195433 RealDigits[%, 10, 100] (* (A) A195433 *)
%t A195433 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195433 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195433 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195433 RealDigits[%, 10, 100] (* (B)=(2/3)sqrt(2); -1+A179587 *)
%t A195433 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195433 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195433 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195433 RealDigits[%, 10, 100] (* (C) A195433 *)
%t A195433 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195433 RealDigits[%, 10, 100] (* Philo(ABC,G) A195436 *)
%Y A195433 Cf. A195304, A195434, A195435, A195436.
%K A195433 nonn,cons
%O A195433 0,1
%A A195433 _Clark Kimberling_, Sep 18 2011