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A195405 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)=(1,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).

%I A195405 #8 May 13 2017 14:18:14
%S A195405 9,2,4,8,7,5,3,9,1,0,5,0,2,2,5,1,3,0,6,6,2,6,2,5,1,7,3,5,1,2,7,4,5,4,
%T A195405 1,0,7,5,2,6,0,3,3,5,1,6,5,1,0,7,9,4,9,3,7,5,4,9,9,2,8,7,4,8,9,5,6,7,
%U A195405 6,4,5,9,7,1,1,9,6,7,4,8,8,3,6,5,6,5,2,1,1,4,4,1,6,1,0,2,5,4,6,0
%N A195405 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)=(1,sqrt(r),r), where r=(1+sqrt(5))/2 (the golden ratio).
%C A195405 See A195284 for definitions and a general discussion.
%e A195405 (C)=0.92487539105022513066262517351274541075260...
%t A195405 a = 1; b = Sqrt[c]; c = (1 + Sqrt[5])/2;
%t A195405 f = 2 a*b/(a + b + c);
%t A195405 x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
%t A195405 x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
%t A195405 x3 = f*Sqrt[2]
%t A195405 N[x1, 100]
%t A195405 RealDigits[%]    (* (A) A195403 *)
%t A195405 N[x2, 100]
%t A195405 RealDigits[%]    (* (B) A195404 *)
%t A195405 N[x3, 100]
%t A195405 RealDigits[%]    (* (C) A195405 *)
%t A195405 N[(x1 + x2 + x3)/(a + b + c), 100]
%t A195405 RealDigits[%]    (*  Philo(ABC,I) A195406 *)
%Y A195405 Cf. A195284.
%K A195405 nonn,cons
%O A195405 0,1
%A A195405 _Clark Kimberling_, Sep 17 2011