Let \({\mathcal A}\) be a regular \(n\)-gon with vertices \(A_ 1,A_ 2,\ldots,A_ n\) and edge-length =1, which is inscribed in \({\mathcal P}\), an \(n\)-gon with vertices \(P_ 1,P_ 2,\ldots,P_ n\), in such a way that \(A_ 1P_ 1=A_ 2P_ 2=\cdots=A_ nP_ n\). The main results of this paper are given in the following theorem. Theorem 1. If \(n=3\) or 4 or \(\geq 7\) and odd, then \({\mathcal P}\) must be regular; while if \(n\geq 6\) is even, then \({\mathcal P}\) may be nonregular; for \(n\geq 8\), \({\mathcal P}\) has only two sizes of angles and these alternate, and the sides of \({\mathcal P}\) also alternate in size.