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A195429 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)=(8,15,17).

%I A195429 #5 Mar 30 2012 18:57:45
%S A195429 5,1,7,3,5,3,1,7,6,9,8,3,7,2,5,8,2,9,8,7,7,0,6,2,9,5,8,5,1,6,9,4,5,9,
%T A195429 7,3,6,9,1,8,7,6,6,2,8,8,3,4,7,7,9,1,4,5,8,0,7,8,6,2,8,2,2,6,5,2,8,7,
%U A195429 0,9,0,5,8,2,6,1,4,0,1,9,2,3,8,7,1,8,8,0,3,9,1,8,1,3,2,8,6,1,8,3
%N A195429 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)=(8,15,17).
%C A195429 See A195304 for definitions and a general discussion.
%e A195429 (A)=5.173531769837258298770629585169459736918766...
%t A195429 a = 8; b = 15; h = 2 a/3; k = b/3;
%t A195429 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195429 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195429 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195429 RealDigits[%, 10, 100] (* (A) A195429 *)
%t A195429 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195429 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195429 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195429 RealDigits[%, 10, 100] (* (B) A195430 *)
%t A195429 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195429 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195429 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195429 RealDigits[%, 10, 100] (* (C) A195431 *)
%t A195429 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195429 RealDigits[%, 10, 100] (* Philo(ABC,G) A195432 *)
%Y A195429 Cf. A195304, A195430, A195431, A195432.
%K A195429 nonn,cons
%O A195429 1,1
%A A195429 _Clark Kimberling_, Sep 18 2011