A360828 - OEIS

A360828

Decimal expansion of the ratio between the perimeter of the first Morley triangle of an isosceles right triangle and the perimeter of this isosceles right triangle.

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

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OFFSET

0,2

COMMENTS

The first Morley triangle, also called the Morley triangle, of any triangle is always equilateral (see Wikipedia link).

If an isosceles right triangle ABC has side lengths (a, a, a*sqrt(2)), then it has a circumradius R = a*sqrt(2)/2, and a perimeter P = (2 + sqrt(2))*a, and its first Morley triangle has side a' and perimeter P' = 3*a', with a' = 8*R*sin(Pi/6)*sin(Pi/12)*sin(Pi/12) = a*sqrt(2)*(2-sqrt(3))/2. This gives the ratio P'/P = (3/2) * (sqrt(2)-1) * (2-sqrt(3)) (see Illustration).

LINKS

Table of n, a(n) for n=0..99.

Bernard Schott, Illustration of the Morley triangle of an isosceles right triangle.

Wikipedia, Morley's trisector theorem.

FORMULA

Equals (3/2) * (sqrt(2)-1) * (2-sqrt(3)).

EXAMPLE

0.1664822842978339394003095728290656981743022858...

MAPLE

evalf((1/2)*(3*(sqrt(2)-1))*(2-sqrt(3)), 100);

MATHEMATICA

RealDigits[(3/2)*(Sqrt[2] - 1)*(2 - Sqrt[3]), 10, 100][[1]] (* Amiram Eldar, Feb 28 2023 *)