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A195288
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Decimal expansion of shortest length, (C), of segment from side CA through incenter to side CB in right triangle ABC with sidelengths (a,b,c)=(5,12,13).
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4
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4, 8, 0, 7, 4, 0, 1, 7, 0, 0, 6, 1, 8, 6, 5, 2, 3, 9, 0, 8, 2, 5, 6, 2, 8, 3, 5, 6, 6, 2, 7, 3, 2, 7, 9, 2, 8, 3, 3, 5, 0, 6, 2, 0, 9, 8, 4, 6, 0, 3, 2, 8, 2, 8, 3, 6, 1, 3, 9, 3, 7, 4, 0, 8, 3, 0, 2, 8, 8, 9, 2, 6, 4, 3, 9, 0, 6, 8, 0, 5, 9, 3, 6, 0, 6, 1, 5, 8, 7, 7, 6, 0, 2, 4, 6, 5, 4, 2, 9, 0
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OFFSET
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1,1
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COMMENTS
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See A195284 for definitions and a general discussion.
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LINKS
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EXAMPLE
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(C)=4.80740170061865239082562835...
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MATHEMATICA
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a = 5; b = 12; c = 13;
h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2;
s = NSolve[D[f[t], t] == 0, t, 150]
f1 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (A) A195286 *)
f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f3 = (f[t])^(1/2) /. Part[s, 1]
RealDigits[%, 10, 100] (* (B) A195288 *)
f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
s = NSolve[D[f[t], t] == 0, t, 150]
f2 = (f[t])^(1/2) /. Part[s, 4]
RealDigits[%, 10, 100] (* (C) A010487 *)
(f1 + f2 + f3)/(a + b + c)
RealDigits[%, 10, 100] (* Philo(A, B, C, I) A195289 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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