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A195487 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)=(sqrt(7),3,4).

%I A195487 #5 Mar 30 2012 18:57:47
%S A195487 1,6,4,8,0,4,0,7,3,4,5,9,5,5,1,8,8,1,2,3,3,0,7,4,0,7,0,0,9,4,8,4,8,8,
%T A195487 9,2,2,2,3,4,2,5,1,2,5,1,9,9,2,0,3,5,7,8,6,0,3,5,7,3,9,0,9,3,4,3,2,9,
%U A195487 9,6,6,9,6,6,4,8,2,3,6,9,4,9,7,1,6,9,3,2,4,3,7,6,2,1,9,6,0,1,1,7
%N A195487 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)=(sqrt(7),3,4).
%C A195487 See A195304 for definitions and a general discussion.
%e A195487 (A)=1.648040734595518812330740700...
%t A195487 a = Sqrt[7]; b = 3; h = 2 a/3; k = b/3;
%t A195487 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195487 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195487 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195487 RealDigits[%, 10, 100] (* (A) A195487 *)
%t A195487 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195487 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195487 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195487 RealDigits[%, 10, 100] (* (B) A195488 *)
%t A195487 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195487 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195487 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195487 RealDigits[%, 10, 100] (* (C) A195489  *)
%t A195487 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195487 RealDigits[%, 10, 100] (* Philo(ABC,G) A195490 *)
%Y A195487 Cf. A195304, A195488, A195489, A195490.
%K A195487 nonn,cons
%O A195487 1,2
%A A195487 _Clark Kimberling_, Sep 19 2011