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%I #10 Mar 30 2012 18:57:45
%S 6,0,6,0,9,1,5,2,6,7,3,1,3,2,6,4,4,9,4,8,6,4,3,8,0,2,4,6,6,1,2,9,9,1,
%T 7,6,5,2,9,8,5,9,3,7,5,1,6,1,5,4,9,1,7,4,2,1,8,5,7,7,0,3,0,5,6,7,4,5,
%U 6,7,7,6,4,8,3,7,6,0,1,5,9,5,0,7,3,0,8,9,4,3,2,8,3,2,7,4,9,5,9,7
%N Decimal expansion of shortest length, (A), of segment from side AB through incenter to side AC in right triangle ABC with sidelengths (a,b,c)=(7,24,25).
%C See A195284 for definitions and a general discussion.
%e (A)=6.0609152673132644948643802466...
%t a = 7; b = 24; c = 25;
%t h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
%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) A195290 *)
%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] (* (B)=7.5 *)
%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] (* (C) A010524 *)
%t (f1 + f2 + f3)/(a + b + c)
%t RealDigits[%, 10, 100] (* Philo(ABC,I) A195292 *)
%Y Cf. A195284, A195292.
%K nonn,cons
%O 1,1
%A _Clark Kimberling_, Sep 14 2011
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