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A195456 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(3),sqrt(5)).

%I A195456 #10 Dec 24 2017 09:26:00
%S A195456 1,0,3,9,0,8,1,4,7,6,5,9,6,9,8,9,2,3,0,5,5,0,3,1,1,6,2,1,1,2,9,7,8,9,
%T A195456 1,0,3,0,4,1,8,7,9,6,3,9,5,4,0,3,7,6,8,1,8,9,3,8,7,8,7,0,8,0,5,9,7,8,
%U A195456 5,4,1,3,5,3,1,5,9,9,7,9,0,3,1,4,5,6,9,5,2,5,5,6,5,2,3,6,1,1,4,7
%N A195456 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(3),sqrt(5)).
%C A195456 See A195304 for definitions and a general discussion.
%H A195456 G. C. Greubel, <a href="/A195456/b195456.txt">Table of n, a(n) for n = 1..10000</a>
%e A195456 (C)=1.03908147659698923055031162112978910304...
%t A195456 a = Sqrt[2]; b = Sqrt[3]; h = 2 a/3; k = b/3;
%t A195456 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195456 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195456 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195456 RealDigits[%, 10, 100] (* (A) A195454 *)
%t A195456 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195456 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195456 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195456 RealDigits[%, 10, 100] (* (B) A195455 *)
%t A195456 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195456 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195456 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195456 RealDigits[%, 10, 100] (* (C) A195456 *)
%t A195456 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195456 RealDigits[%, 10, 100] (* Philo(ABC,G) A195457 *)
%Y A195456 Cf. A195304, A195454, A195455, A195457.
%K A195456 nonn,cons
%O A195456 1,3
%A A195456 _Clark Kimberling_, Sep 19 2011