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A195425 Decimal expansion of shortest length, (A), of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(7,24,25).

%I A195425 #5 Mar 30 2012 18:57:45
%S A195425 4,6,1,9,7,4,5,4,3,9,8,3,9,2,0,2,0,4,1,2,3,3,4,1,3,5,5,1,1,2,8,8,7,5,
%T A195425 1,3,2,9,8,3,3,3,9,2,9,7,2,3,9,9,1,7,2,4,3,7,1,4,5,8,3,1,5,5,8,0,6,0,
%U A195425 9,4,1,2,3,1,3,1,7,1,1,6,7,7,8,1,2,3,0,2,3,2,0,7,4,5,6,0,8,8,1,2
%N A195425 Decimal expansion of shortest length, (A), of segment from side AB through centroid to side AC in right triangle ABC with sidelengths (a,b,c)=(7,24,25).
%C A195425 See A195304 for definitions and a general discussion.
%e A195425 (A)=4.6197454398392020412334135511288...
%t A195425 a = 7; b = 24; h = 2 a/3; k = b/3;
%t A195425 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195425 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195425 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195425 RealDigits[%, 10, 100] (* (A) A195425 *)
%t A195425 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195425 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195425 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195425 RealDigits[%, 10, 100] (* (B) A195426 *)
%t A195425 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195425 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195425 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195425 RealDigits[%, 10, 100] (* (C) A195427 *)
%t A195425 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195425 RealDigits[%, 10, 100] (* Philo(ABC,G) A195428 *)
%Y A195425 Cf. A195304, A195426, A195427, A195428.
%K A195425 nonn,cons
%O A195425 1,1
%A A195425 _Clark Kimberling_, Sep 18 2011