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A195407 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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

%I A195407 #8 May 13 2017 14:24:20
%S A195407 5,1,2,5,2,2,2,7,2,3,6,2,2,2,5,3,7,9,2,6,3,5,4,9,4,5,5,8,1,0,7,3,5,5,
%T A195407 1,6,9,4,7,8,2,1,9,0,8,2,6,1,4,2,4,2,5,7,4,2,0,1,3,0,4,2,4,3,2,2,0,8,
%U A195407 9,6,5,5,7,2,5,0,5,7,7,4,0,5,1,8,9,2,2,1,3,7,8,5,6,1,3,0,7,0,5,9
%N A195407 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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).
%C A195407 See A195284 for definitions and a general discussion.
%e A195407 (A)=0.51252227236222537926354945581073551694...
%t A195407 a = b - 1; b = (1 + Sqrt[5])/2; c = Sqrt[3];
%t A195407 f = 2 a*b/(a + b + c);
%t A195407 x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
%t A195407 x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
%t A195407 x3 = f*Sqrt[2]
%t A195407 N[x1, 100]
%t A195407 RealDigits[%]    (* (A) A195407 *)
%t A195407 N[x2, 100]
%t A195407 RealDigits[%]    (* (B) A195408 *)
%t A195407 N[x3, 100]
%t A195407 RealDigits[%]    (* (C) A195409 *)
%t A195407 N[(x1 + x2 + x3)/(a + b + c), 100]
%t A195407 RealDigits[%]    (*  Philo(ABC,I) A195410 *)
%Y A195407 Cf. A195284, A195408, A195409, A195410.
%K A195407 nonn,cons
%O A195407 0,1
%A A195407 _Clark Kimberling_, Sep 17 2011