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Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(1,2,sqrt(5)).
6

%I #5 Mar 30 2012 18:57:45

%S 1,3,8,7,3,1,2,7,2,8,3,1,3,8,2,0,9,1,7,4,6,3,3,6,0,2,4,0,9,8,2,2,3,3,

%T 2,1,2,5,9,6,4,4,1,8,6,2,5,1,6,9,7,6,7,6,7,4,7,6,5,4,1,2,8,4,2,3,5,6,

%U 2,8,3,4,5,5,0,0,9,7,1,9,7,9,4,1,5,3,7,9,6,0,7,3,5,5,9,4,5,3,7,4

%N Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(1,2,sqrt(5)).

%C See A195304 for definitions and a general discussion.

%e (B)=1.387312728313820917463360240982233212...

%t a = 1; b = 2; h = 2 a/3; k = b/3;

%t f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2

%t s = NSolve[D[f[t], t] == 0, t, 150]

%t f1 = (f[t])^(1/2) /. Part[s, 4]

%t RealDigits[%, 10, 100] (* (A) A195434 *)

%t f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2

%t s = NSolve[D[f[t], t] == 0, t, 150]

%t f2 = (f[t])^(1/2) /. Part[s, 4]

%t RealDigits[%, 10, 100] (* (B) A195435 *)

%t f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2

%t s = NSolve[D[f[t], t] == 0, t, 150]

%t f3 = (f[t])^(1/2) /. Part[s, 1]

%t RealDigits[%, 10, 100] (* (C) A195444 *)

%t c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)

%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195445 *)

%Y Cf. A195304.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Sep 18 2011