Transforming Polygons via Vertex-Centroid Moves

Given an arbitrary polygon, transform it by a finite sequence of \textquotedblleft{}vertex-centroid\textquotedblright{} moves to a regular polygon. A vertex-centroid move is a translation of a vertex $v$ along the line $vm$, where $m$ is the centroid of the vertices of the polygon, i.e., $1/n$-th of the sum of the vertex coordinates. Vertices may move only one at a time, but in any order and any number of times.