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A195485 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)=(sqrt(2),sqrt(5),sqrt(7)).

%I A195485 #8 Jan 27 2018 02:29:34
%S A195485 1,2,9,4,2,3,8,9,2,3,6,9,2,2,7,3,8,7,4,3,3,4,5,6,7,8,9,9,6,5,6,5,5,0,
%T A195485 5,9,4,6,4,0,8,1,9,5,8,2,9,5,1,9,7,0,1,8,3,0,3,2,9,5,3,4,0,2,4,7,2,2,
%U A195485 1,7,9,1,1,7,9,0,2,0,9,5,3,6,0,0,2,8,4,7,7,3,2,3,6,3,9,2,3,2,6,3
%N A195485 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)=(sqrt(2),sqrt(5),sqrt(7)).
%C A195485 See A195304 for definitions and a general discussion.
%H A195485 G. C. Greubel, <a href="/A195485/b195485.txt">Table of n, a(n) for n = 1..10000</a>
%e A195485 (C)=1.294238923692273874334567899656550594640819...
%t A195485 a = Sqrt[2]; b = Sqrt[5]; h = 2 a/3; k = b/3;
%t A195485 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195485 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195485 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195485 RealDigits[%, 10, 100] (* (A) A195483 *)
%t A195485 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195485 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195485 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195485 RealDigits[%, 10, 100] (* (B) A195484 *)
%t A195485 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195485 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195485 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195485 RealDigits[%, 10, 100] (* (C) A195485 *)
%t A195485 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195485 RealDigits[%, 10, 100] (* Philo(ABC,G) A195486 *)
%Y A195485 Cf. A195304.
%K A195485 nonn,cons
%O A195485 1,2
%A A195485 _Clark Kimberling_, Sep 19 2011