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%I #5 Mar 30 2012 18:57:45
%S 1,3,8,2,1,1,5,8,6,0,2,8,3,0,9,9,8,8,2,6,2,0,7,5,3,9,9,1,3,0,7,2,8,0,
%T 2,7,9,6,5,1,8,4,5,0,4,8,2,3,5,7,9,5,9,2,6,9,4,3,5,8,5,0,5,0,6,0,8,5,
%U 3,2,2,1,3,5,1,9,2,4,4,8,4,5,0,1,0,0,3,8,2,2,7,4,4,9,2,4,2,6,4,6
%N 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)=(7,24,25).
%C See A195304 for definitions and a general discussion.
%e (B)=13.821158602830998826207539913072802796518450...
%t a = 7; b = 24; 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) A195425 *)
%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) A195426 *)
%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) A195427 *)
%t c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t RealDigits[%, 10, 100] (* Philo(ABC,G) A195428 *)
%Y Cf. A195304.
%K nonn,cons
%O 2,2
%A _Clark Kimberling_, Sep 18 2011
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