Let \(D\) be a Jordan domain in \(\overline \mathbb{R}^2\). We say that \(D\) is a quasiconvex domain if there exists a constant a such that each pair of points \(x_1\), \(x_2\in D\) can be joined by a rectifiable curve \(\gamma\) in \(D\) satisfying \(1(\gamma) \leq a |x_1-x_2 |\), where \(1(\gamma)\) is the Euclidean length of the curve \(\gamma\), and that \(D\) is a cigar domain if there exists \(a\) constant \(b\) such that \(\min_{j=1,2} 1(\gamma (x_j,x))b\) \(d(x, \partial D)\) for any \(x\in \gamma\), where \(\gamma (x_1,x)\) and \(\gamma (x_2,x)\) are the two components of \(\gamma \setminus \{x\}\) and \(d(x, \partial D)\) is the distance between the point \(x\) and the boundary of the domain \(D\). Suppose \(D^* =\overline R^2 \setminus D\). In this paper, the authors prove the following results: Theorem 1. \(D\) is a quasidisk if and only if both \(D\) and \(D^*\) are quasiconvex domains. Theorem 2. \(D\) is a quasidisk if and only if both \(D\) and \(D^*\) are cigar domains.