Let $K$ be a quadratic field and $\mathfrak{a}$ be a fixed integral ideal of $O_K$. In this paper, we investigate the distribution of ideals that divide $\mathfrak{a}$ using the Selberg-Delange method. This is a natural variation of a result studied by Deshouillers, Dress, and Tenenbaum (often referred to as the DDT Theorem), and we find that this distribution converges to the arcsine distribution.