Some characterizations of the circle related to circumscribed equiangular polygons

The main purpose of this note is to prove the following: let n ≥ 3 be a fixed integer number and let K be a convex body such that for every equiangular circumscribed n -gon, the midpoints of its sides belong to K . Then K is a disc. We also prove that a 3-dimensional convex body K such that the centers of the faces of all its circumscribed regular tetrahedra belong to K is a ball.