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A195290 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). 3

%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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Last modified March 28 08:00 EDT 2024. Contains 371235 sequences. (Running on oeis4.)