A359837 Decimal expansion of the unsigned ratio of similitude between an equilateral reference triangle and its first Morley triangle.

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

Offset: 0

Author

Frank M Jackson, Jan 14 2023

Keywords

• easy
• nonn
• cons

Comments

The first Morley triangle of any reference triangle is always equilateral. Therefore a reference equilateral triangle and its first Morley triangle will be in a homothetic relationship. This sequence is the real terms of a constant that is the ratio of similitude of the homothety. The constant is negative.

If an equilateral triangle has a side a, a circumradius R and a first Morley triangle with side a', then a = R*sqrt(3) and a' = 8*R*(sin(Pi/9))^3, so the ratio of similitude between the two triangles is a'/a = (8/sqrt(3)) * (sin(Pi/9))^3. - Bernard Schott, Feb 06 2023

Examples

			0.1847925309040953727013520475722037558709135597926517172356...
		

Links

• Wikipedia, Morley's trisector theorem.
• Index entries for algebraic numbers, degree 3.

Crossrefs

Cf. A019819, A019829, A019879, A020760.

Programs

Mathematica

RealDigits[Sin[Pi/18]/Cos[Pi/9], 10, 100][[1]]
N[Solve[x^3 + 3*x^2 - 6*x + 1 == 0, {x}][[2]], 90]

PARI

sin(Pi/18)/cos(Pi/9) \\ Michel Marcus, Jan 15 2023

Formula

Equals sin(Pi/18)/cos(Pi/9).

A root of x^3+3*x^2-6*x+1.

Equals A019819/A019879. - Michel Marcus, Jan 15 2023

Equals 8 * A020760 * A019829^3. - Bernard Schott, Feb 06 2023