Problem of Cutting a Polygon into Two Congruent Parts

In the paper, we discuss a version of the classical problem of partitioning a figure into two congruent parts, which was posed in 1999. We consider the conjecture that when partitioning a convex centrally symmetric figure into two congruent parts, the center of symmetry always lies on the common boundary of the two parts. The main result of the paper consists in proving the conjecture for simple but not necessarily convex figures on the plane under the condition that the figure is partitioned into two parts by a curve that is not self-intersecting. There is a certain advance in classifying the types of motions in the general case on the plane.