Show that the marked region is an equilateral triangle



The picture is symmetric with respect to the line going from the midpoints of two opposite sides of the square and passing through \(A\).
Therefore, the triangle is symmetric with respect to this line; hence, two sides of the marked triangle are equal.
The angle at \(A\) formed by these two sides equals \(60^\circ\), because the triangle shares this angle with an equilateral triangle.
Hence the marked triangle is an equilateral triangle.