Show that the marked quadrilateral is a square



The diagonals of the marked quadrilateral are diameters of the circle.
They are constructed by two heights of the equilateral triangle and the side length of the square.
As the heights of the equilateral triangles are perpendicular to the square's sides, we can conclude that the diagonals of the marked quadrilateral are perpendicular.
Hence, the diagonals of the marked quadrilateral are equal and perpendicular.
Thus, it is a square.