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A195413 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).

%I A195413 #5 Mar 30 2012 18:57:45
%S A195413 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 A195413 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 A195413 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 A195413 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 A195413 See A195304 for definitions and a general discussion.
%e A195413 (B)=7.77773177751211566868403389221547...
%t A195413 a = 5; b = 12; h = 2 a/3; k = b/3;
%t A195413 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195413 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195413 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195413 RealDigits[%, 10, 100] (* (A) A195412 *)
%t A195413 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195413 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195413 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195413 RealDigits[%, 10, 100] (* (B) A195413 *)
%t A195413 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195413 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195413 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195413 RealDigits[%, 10, 100] (* (C) A195414 *)
%t A195413 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195413 RealDigits[%, 10, 100] (* Philo(ABC,G) A195424 *)
%Y A195413 Cf. A195304.
%K A195413 nonn,cons
%O A195413 1,1
%A A195413 _Clark Kimberling_, Sep 18 2011