A360829 - OEIS

%I #25 Feb 05 2025 10:04:32

%S 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,

%T 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,

%U 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

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

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

%C 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).

%C 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.

%H Bernard Schott, <a href="/A360829/a360829.pdf">Illustration of the Morley triangle of an isosceles right triangle</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Morley&#39;s_trisector_theorem">Morley's trisector theorem</a>.

%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.

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

%e 0.03108891324553526367303109763527664214990919416...

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

%t RealDigits[(7*Sqrt[3] - 12)/4, 10, 100][[1]] (* _Amiram Eldar_, Mar 09 2023 *)

%o (PARI) 7*sqrt(3)/4 - 3 \\ _Charles R Greathouse IV_, Feb 05 2025

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

%K nonn,cons,changed

%O -1,1

%A _Bernard Schott_, Mar 09 2023