A360829 Decimal expansion of the ratio between the area of the first Morley triangle of an isosceles right triangle and its area.

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

Offset

-1,1

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 an area A = a^2/2, and its first Morley triangle has side a' = 8*R*sin(Pi/6)*sin(Pi/12)*sin(Pi/12) and an area A' = a'^2 * sqrt(3)/4 = a^2 * (7*sqrt(3) - 12)/8. This gives the ratio A'/A = (7*sqrt(3)-12)/4 (see Illustration).

This ratio is not equal to the square of the ratio of the perimeters = A360828^2 because the Morley triangle and the isosceles right triangle are not homothetic.

Links

Table of n, a(n) for n=-1..96.

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

Wikipedia, Morley's trisector theorem.

Formula

Equals (7*sqrt(3) - 12)/4.

Example

0.03108891324553526367303109763527664214990919416...

Maple

evalf((7/4)*sqrt(3) - 3, 100);

Mathematica

RealDigits[(7*Sqrt[3] - 12)/4, 10, 100][[1]] (* Amiram Eldar, Mar 09 2023 *)

Crossrefs

Cf. A359837 (ratio of perimeters in the case of an equilateral triangle), A360828 (ratio of perimeters in the case of an isosceles right triangle).

Sequence in context: A052420 A348096 A162971 * A078521 A194938 A299614

Adjacent sequences: A360826 A360827 A360828 * A360830 A360831 A360832

Keyword

nonn,cons

Author

Bernard Schott, Mar 09 2023

Status

approved