Medieval stonemasons used this method to construct exact octagons in a given square window.

Open your compasses to a radius of half the diagonal of the square and construct an arc with centre one vertex of the square - mark the 2 points where the arc crosses the sides.

Do that for all 4 vertices of the square giving 8 points which are the vertices of an octagon. Is the octagon an exact regular octagon ? Proof please.