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A195473 Decimal expansion of shortest length, (C), of segment from side CA through centroid to side CB in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(2),sqrt(3)).

%I A195473 #8 Jan 27 2018 02:29:25
%S A195473 8,3,1,9,7,7,5,6,0,2,8,9,1,6,3,2,0,4,5,9,3,0,2,3,8,1,1,4,8,1,9,6,7,8,
%T A195473 2,7,4,4,1,2,5,0,3,0,4,9,9,1,9,8,6,7,8,3,5,4,9,3,4,1,1,3,7,0,4,5,9,1,
%U A195473 4,2,8,7,4,9,7,7,6,9,9,2,5,9,7,0,5,8,3,3,2,4,3,6,9,8,7,6,3,7,8,7
%N A195473 Decimal expansion of shortest length, (C), of segment from side CA through centroid to side CB in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(2),sqrt(3)).
%C A195473 See A195304 for definitions and a general discussion.
%H A195473 G. C. Greubel, <a href="/A195473/b195473.txt">Table of n, a(n) for n = 0..10000</a>
%e A195473 (C)=0.8319775602891632045930238114819678...
%t A195473 a = 1; b = Sqrt[2]; h = 2 a/3; k = b/3;
%t A195473 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195473 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195473 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195473 RealDigits[%, 10, 100] (* (A) A195471 *)
%t A195473 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195473 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195473 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195473 RealDigits[%, 10, 100] (* (B) A195472 *)
%t A195473 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195473 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195473 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195473 RealDigits[%, 10, 100] (* (C) A195473 *)
%t A195473 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195473 RealDigits[%, 10, 100] (* Philo(ABC,G) A195474 *)
%Y A195473 Cf. A195304.
%K A195473 nonn,cons
%O A195473 0,1
%A A195473 _Clark Kimberling_, Sep 19 2011