Show that the marked octagon is a regular octagon



Consider the square and the four quarters of circle.
This 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.
We deduce that the octagon also has these four symmetry axes hence all of its interior angles are the same.

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 and .