A360829 - OEIS

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, 7, 4

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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..97.

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

Wikipedia, Morley's trisector theorem.

Index entries for algebraic numbers, degree 2.

FORMULA

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

EXAMPLE

0.03108891324553526367303109763527664214990919416...

MAPLE

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

MATHEMATICA