A195381 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)=(2,sqrt(5),3).

1, 3, 2, 3, 1, 6, 9, 0, 7, 6, 4, 9, 9, 2, 1, 4, 9, 9, 5, 4, 0, 3, 0, 7, 3, 6, 2, 4, 7, 3, 5, 2, 1, 7, 4, 8, 9, 9, 9, 5, 4, 9, 4, 0, 5, 6, 1, 3, 9, 5, 5, 1, 0, 5, 7, 5, 7, 9, 8, 4, 7, 1, 7, 2, 2, 4, 2, 3, 1, 5, 9, 5, 8, 7, 8, 9, 4, 2, 1, 0, 7, 7, 7, 2, 4, 1, 5, 1, 1, 8, 3, 4, 1, 3, 0, 7, 2, 2, 0, 9

Offset: 1

Author

Clark Kimberling, Sep 17 2011

Keywords

• nonn
• cons

Comments

See A195284 for definitions and a general discussion.

Examples

			(A)=1.32316907649921499540307362473521748999...
		

Crossrefs

Cf. A195284, A195383, A195384, A195385.

Programs

Magma

Sqrt(12) / ((1 + Sqrt(5)) / 2)^2; // Vincenzo Librandi, Nov 15 2018

Mathematica

a = 2; b = Sqrt[5]; c = 3; f = 2 a*b/(a + b + c);
x1 = Simplify[f*Sqrt[a^2 + (b + c)^2]/(b + c) ]
x2 = Simplify[f*Sqrt[b^2 + (c + a)^2]/(c + a) ]
x3 = f*Sqrt[2]
N[x1, 100]
RealDigits[%]   (* (A) A195381 *)
N[x2, 100]
RealDigits[%]   (* (B) A195383 *)
N[x3, 100]
RealDigits[%]   (* (C) A195384 *)
N[(x1 + x2 + x3)/(a + b + c), 100]
RealDigits[%]   (* Philo(ABC,I) A195385 *)
RealDigits[Sqrt[12] / ((1 + Sqrt[5]) / 2)^2, 10, 100] (* Vincenzo Librandi, Nov 15 2018 *)

Formula

Equals sqrt(12)/phi^2, where phi = A001622. - Jon Maiga, Nov 14 2018