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A195298 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)=(28,45,53).

%I A195298 #9 Mar 30 2012 18:57:45
%S A195298 2,0,8,0,0,3,1,3,9,6,9,3,7,2,9,0,9,3,4,5,9,9,2,2,9,2,8,3,2,9,3,4,3,7,
%T A195298 9,4,1,0,7,9,3,3,4,1,9,5,0,2,1,8,5,0,6,9,6,6,7,9,4,8,0,5,1,1,7,9,5,4,
%U A195298 6,1,6,3,9,6,0,7,1,1,5,7,6,6,6,6,5,5,9,4,1,1,6,8,8,0,2,6,4,7,8
%N A195298 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)=(28,45,53).
%C A195298 See A195284 for definitions and a general discussion.
%e A195298 (A)=20.800313969372909345992292832934379410...
%t A195298 a = 28; b = 45; c = 53;
%t A195298 h = a (a + c)/(a + b + c); k = a*b/(a + b + c);
%t A195298 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2;
%t A195298 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195298 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195298 RealDigits[%, 10, 100] (* (A) A195298 *)
%t A195298 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195298 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195298 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195298 RealDigits[%, 10, 100] (* (B) A195299 *)
%t A195298 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195298 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195298 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195298 RealDigits[%, 10, 100] (* (C)=20*sqrt(2) *)
%t A195298 (f1 + f2 + f3)/(a + b + c)
%t A195298 RealDigits[%, 10, 100]  (* Phil(ABC,I), A195300 *)
%Y A195298 Cf. A195284, A195299, A195300, A195301.
%K A195298 nonn,cons
%O A195298 2,1
%A A195298 _Clark Kimberling_, Sep 14 2011