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A360828
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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.
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2
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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
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0,2
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COMMENTS
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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).
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LINKS
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FORMULA
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Equals (3/2) * (sqrt(2)-1) * (2-sqrt(3)).
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EXAMPLE
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0.1664822842978339394003095728290656981743022858...
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MAPLE
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evalf((1/2)*(3*(sqrt(2)-1))*(2-sqrt(3)), 100);
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MATHEMATICA
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RealDigits[(3/2)*(Sqrt[2] - 1)*(2 - Sqrt[3]), 10, 100][[1]] (* Amiram Eldar, Feb 28 2023 *)
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PROG
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(PARI) (3/2) * (sqrt(2)-1) * (2-sqrt(3)) \\ Michel Marcus, Mar 03 2023
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CROSSREFS
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Cf. A359837 (ratio of perimeters in the case of an equilateral triangle), A360829 (ratio of areas in the case of an isosceles right triangle).
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KEYWORD
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AUTHOR
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STATUS
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approved
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