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Choose any four points on a circle and join them to form a quadrilateral. Drawing the diagonals of this quadrilateral produces four overlapping triangles (each diagonal creates two of them).

Draw the largest possible circle in each of these triangles, and the centers of these circles will always form a rectangle:

from Futility Closet https://www.futilitycloset.com/2018/11/08/a-hidden-door/

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