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A195301 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)=(1,1,sqrt(2)). 4

%I

%S 6,3,4,0,5,0,6,7,1,1,2,4,4,2,8,8,5,0,6,8,5,0,5,2,8,8,5,3,4,3,9,6,2,2,

%T 1,3,1,9,8,9,1,0,0,0,3,5,6,9,5,5,3,6,1,2,9,8,9,9,8,5,8,4,0,1,0,1,7,7,

%U 1,7,5,8,3,2,3,6,9,1,8,9,6,9,6,3,2,4,9,4,5,6,6,6,3,1,1,0,0,0

%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)=(1,1,sqrt(2)).

%C See A195284 for definitions and a general discussion.

%H G. C. Greubel, <a href="/A195301/b195301.txt">Table of n, a(n) for n = 0..10000</a>

%e (A)=0.63405067112442885068505288534396221319891000...

%t a = 1; b = 1; c = Sqrt[2];

%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, 1]

%t RealDigits[%, 10, 100] (* (A) A195301 *)

%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, 4]

%t RealDigits[%, 10, 100] (* (B)=(A) *)

%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, 1]

%t RealDigits[%, 10, 100] (* (C) A163960 *)

%t (f1 + f2 + f3)/(a + b + c)

%t RealDigits[%, 10, 100] (* Philo(ABC,I), A195303 *)

%Y Cf. A195284, A195303, A195304.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Sep 14 2011

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Last modified June 14 09:06 EDT 2021. Contains 345018 sequences. (Running on oeis4.)