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A195409 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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).

%I A195409 #10 Jul 18 2021 10:02:11
%S A195409 7,1,2,7,8,7,9,1,7,3,8,5,2,0,1,2,3,3,8,0,1,6,0,9,4,6,9,7,2,6,8,2,7,1,
%T A195409 4,1,7,5,3,6,0,7,6,5,8,6,6,8,5,4,6,6,9,8,4,2,4,8,1,2,2,8,5,5,4,1,6,3,
%U A195409 4,0,6,1,1,8,1,9,2,3,1,9,4,8,0,4,3,8,8,6,7,5,2,7,4,6,6,0,0,6,0,3,6,8,7,5
%N A195409 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)=(r-1,r,sqrt(3)), where r=(1+sqrt(5))/2 (the golden ratio).
%C A195409 See A195284 for definitions and a general discussion.
%e A195409 (C)=0.71278791738520123380160946972682714175360765866...
%t A195409 a = b - 1; b = (1 + Sqrt[5])/2; c = Sqrt[3];
%t A195409 f = 2 a*b/(a + b + c);
%t A195409 x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
%t A195409 x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
%t A195409 x3 = f*Sqrt[2]
%t A195409 N[x1, 100]
%t A195409 RealDigits[%]    (* (A) A195407 *)
%t A195409 N[x2, 100]
%t A195409 RealDigits[%]    (* (B) A195408 *)
%t A195409 N[x3, 100]
%t A195409 RealDigits[%]    (* (C) A195409 *)
%t A195409 N[(x1 + x2 + x3)/(a + b + c), 100]
%t A195409 RealDigits[%]    (*  Philo(ABC,I) A195410 *)
%Y A195409 Cf. A195284.
%K A195409 nonn,cons
%O A195409 0,1
%A A195409 _Clark Kimberling_, Sep 17 2011
%E A195409 a(99) corrected by _Georg Fischer_, Jul 18 2021