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A195299 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)=(28,45,53).

%I A195299 #8 Mar 30 2012 18:57:45
%S A195299 2,2,8,7,9,1,7,8,0,9,1,0,8,2,2,2,2,9,2,3,9,9,4,1,5,4,3,6,4,8,8,3,4,4,
%T A195299 4,3,9,7,1,0,8,4,4,7,6,0,7,7,5,9,9,0,4,2,7,1,6,5,4,6,8,0,0,9,1,9,9,5,
%U A195299 6,9,3,6,1,7,7,7,2,8,6,3,9,4,2,2,8,7,8,9,5,5,8,5,2,3,9,0,3,4,6
%N A195299 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)=(28,45,53).
%C A195299 See A195284 for definitions and a general discussion.
%e A195299 (C)=
%t A195299 a = 28; b = 45; c = 53;
%t A195299 h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
%t A195299 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2;
%t A195299 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195299 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195299 RealDigits[%, 10, 100] (* (A) A195298 *)
%t A195299 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195299 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195299 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195299 RealDigits[%, 10, 100] (* (B) A195299 *)
%t A195299 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195299 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195299 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195299 RealDigits[%, 10, 100] (* (C)=20*sqrt(2) *)
%t A195299 (f1 + f2 + f3)/(a + b + c)
%t A195299 RealDigits[%, 10, 100]  (* Phil(ABC,I), A195300 *)
%Y A195299 Cf. A195284.
%K A195299 nonn,cons
%O A195299 2,1
%A A195299 _Clark Kimberling_, Sep 14 2011