On relatively equilateral polygons inscribed in a convex body

Let C subset of E-2 be a convex body. The C-length of a segment is the ratio of its length to the half of the length of a longest parallel chord of C. By a relatively equilateral polygon inscribed in C we mean an inscribed convex polygon all of whose sides are of equal C-length. We prove that for every boundary point x of C and every integer k greater than or equal to 3 there exists a relatively equilateral k-gon with vertex x inscribed in C. We discuss the C-length of sides of relatively equilateral k-gons inscribed in C and we reformulate this question in terms of packing C by k homothetical copies which touch the boundary of C.