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A195403 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,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

%I A195403 #10 Jul 18 2021 09:53:56
%S A195403 6,9,2,0,2,8,6,7,8,4,7,1,6,5,1,7,6,7,9,0,4,3,2,8,7,4,5,2,5,6,2,9,3,2,
%T A195403 5,2,0,0,9,4,0,2,2,7,5,9,3,1,3,3,3,2,2,7,0,3,7,6,1,6,4,8,0,3,3,1,9,2,
%U A195403 5,7,7,4,5,6,5,6,6,8,8,7,5,7,5,3,6,4,5,9,7,8,4,0,1,8,6,1,7,5,7,8,9
%N A195403 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,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).
%C A195403 See A195284 for definitions and a general discussion.
%e A195403 (A)=0.69202867847165176790432874525629325200...
%t A195403 a = 1; b = Sqrt[c]; c = (1 + Sqrt[5])/2;
%t A195403 f = 2 a*b/(a + b + c);
%t A195403 x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
%t A195403 x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
%t A195403 x3 = f*Sqrt[2]
%t A195403 N[x1, 100]
%t A195403 RealDigits[%]    (* (A) A195403 *)
%t A195403 N[x2, 100]
%t A195403 RealDigits[%]    (* (B) A195404 *)
%t A195403 N[x3, 100]
%t A195403 RealDigits[%]    (* (C) A195405 *)
%t A195403 N[(x1 + x2 + x3)/(a + b + c), 100]
%t A195403 RealDigits[%]    (*  Philo(ABC,I) A195406 *)
%Y A195403 Cf. A195284, A195404, A195405, A195406.
%K A195403 nonn,cons
%O A195403 0,1
%A A195403 _Clark Kimberling_, Sep 17 2011
%E A195403 a(99) corrected by _Georg Fischer_, Jul 18 2021