Show that the marked octagon is a regular octagon



The figure has four symmetry axes: the two lines containing the square diagonals;
the two lines through the center of the square that are parallel to some of the square sides.
The octagon has these four symmetry axes hence all of its angles are equal.
We now prove that all octagon sides have the same length.
The octagon sides that are contained in square sides have length \(\sqrt{2}-1\) ,see .
The octagon sides that are not contained in square sides have also length \(\sqrt{2}-1\).
Indeed, they are diagonals in a square of side \(1-\frac{\sqrt{2}}{2}\), see .