A195413 - OEIS

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

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

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

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

%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)=(5,12,13).

%C See A195304 for definitions and a general discussion.

%e (B)=7.77773177751211566868403389221547...

%t a = 5; b = 12; 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) A195412 *)

%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) A195413 *)

%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) A195414 *)

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

%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195424 *)

%Y Cf. A195304.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Sep 18 2011