Let $K$ be an imaginary quadratic field and $O_K$ be its ring of integers. We say $O_K$ is norm Euclidean if the norm is a [Euclidean function. ][1] It is known from the classification of imaginary quadratic fields with class number 1 that $O_K$ is Euclidean if and only if it is norm Euclidean. Is there a more straightforward proof of this fact?


  [1]: https://en.wikipedia.org/wiki/Euclidean_domain#Definition