A195407 - OEIS

%I #8 May 13 2017 14:24:20

%S 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 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 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 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 See A195284 for definitions and a general discussion.

%e (A)=0.51252227236222537926354945581073551694...

%t a = b - 1; b = (1 + Sqrt[5])/2; c = Sqrt[3];

%t f = 2 a*b/(a + b + c);

%t x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]

%t x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]

%t x3 = f*Sqrt[2]

%t N[x1, 100]

%t RealDigits[%]    (* (A) A195407 *)

%t N[x2, 100]

%t RealDigits[%]    (* (B) A195408 *)

%t N[x3, 100]

%t RealDigits[%]    (* (C) A195409 *)

%t N[(x1 + x2 + x3)/(a + b + c), 100]

%t RealDigits[%]    (*  Philo(ABC,I) A195410 *)

%Y Cf. A195284, A195408, A195409, A195410.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Sep 17 2011