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A195404 Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

%I A195404 #10 Jul 18 2021 09:56:46
%S A195404 7,2,7,0,9,2,0,6,2,9,2,8,0,7,0,1,2,0,5,2,4,5,5,7,2,3,6,3,8,0,5,8,0,9,
%T A195404 4,1,6,2,4,2,4,2,5,2,1,7,4,5,8,0,8,3,2,5,7,3,6,5,7,5,7,6,7,7,6,9,1,4,
%U A195404 1,5,2,5,3,8,2,8,6,6,1,4,9,5,9,7,1,7,4,1,8,1,0,0,0,1,3,4,3,4,7,4,5,5,9,6,5
%N A195404 Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(1,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).
%C A195404 See A195284 for definitions and a general discussion.
%e A195404 (B)=0.72709206292807012052455723638058094...
%t A195404 a = 1; b = Sqrt[c]; c = (1 + Sqrt[5])/2;
%t A195404 f = 2 a*b/(a + b + c);
%t A195404 x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
%t A195404 x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
%t A195404 x3 = f*Sqrt[2]
%t A195404 N[x1, 100]
%t A195404 RealDigits[%]    (* (A) A195403 *)
%t A195404 N[x2, 100]
%t A195404 RealDigits[%]    (* (B) A195404 *)
%t A195404 N[x3, 100]
%t A195404 RealDigits[%]    (* (C) A195405 *)
%t A195404 N[(x1 + x2 + x3)/(a + b + c), 100]
%t A195404 RealDigits[%]    (*  Philo(ABC,I) A195406 *)
%Y A195404 Cf. A195284.
%K A195404 nonn,cons
%O A195404 0,1
%A A195404 _Clark Kimberling_, Sep 17 2011
%E A195404 a(99) corrected by _Georg Fischer_, Jul 18 2021