A195447 - OEIS

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

%S 1,8,0,1,8,6,6,2,4,2,4,7,4,4,9,4,6,0,8,4,7,6,8,6,9,5,5,0,8,9,8,5,4,1,

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

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

%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)=(1,3,sqrt(10)).

%C See A195304 for definitions and a general discussion.

%e (B)=1.8018662424744946084768695508...

%t a = 1; b = 3; 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) A195446  *)

%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) A195447 *)

%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) A195448 *)

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

%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195449 *)

%Y Cf. A195304.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Sep 18 2011