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%I #37 Mar 10 2023 03:38:03
%S 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,
%T 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,
%U 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
%N 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.
%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 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).
%H Bernard Schott, <a href="/A360828/a360828_1.pdf">Illustration of the Morley triangle of an isosceles right triangle</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Morley's_trisector_theorem">Morley's trisector theorem</a>.
%F Equals (3/2) * (sqrt(2)-1) * (2-sqrt(3)).
%e 0.1664822842978339394003095728290656981743022858...
%p evalf((1/2)*(3*(sqrt(2)-1))*(2-sqrt(3)), 100);
%t RealDigits[(3/2)*(Sqrt[2] - 1)*(2 - Sqrt[3]), 10, 100][[1]] (* _Amiram Eldar_, Feb 28 2023 *)
%o (PARI) (3/2) * (sqrt(2)-1) * (2-sqrt(3)) \\ _Michel Marcus_, Mar 03 2023
%Y Cf. A359837 (ratio of perimeters in the case of an equilateral triangle), A360829 (ratio of areas in the case of an isosceles right triangle).
%K nonn,cons
%O 0,2
%A _Bernard Schott_, Feb 28 2023
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