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%I #5 Mar 30 2012 18:57:45
%S 8,4,5,5,7,1,8,1,7,5,1,2,6,5,2,7,6,2,6,9,2,1,0,8,8,0,8,0,5,4,2,1,9,2,
%T 0,8,2,2,4,0,4,3,6,0,1,4,3,4,1,9,9,2,7,6,2,8,4,5,8,1,0,0,3,1,4,5,7,9,
%U 6,3,5,7,2,1,3,9,1,2,0,9,7,2,3,0,7,6,9,8,2,9,1,1,1,6,1,9,2,3,6,4
%N Decimal expansion of shortest length, (C), of segment from side CA through centroid to side CB in right triangle ABC with sidelengths (a,b,c)=(8,15,17).
%C See A195304 for definitions and a general discussion.
%e (C)=8.455718175126527626921088080542192082240...
%t a = 8; b = 15; 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) A195429 *)
%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) A195430 *)
%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) A195431 *)
%t c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195432 *)
%Y Cf. A195304.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Sep 18 2011
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