Why is euthanasia not permitted in Judaism?

This very basic question requires a longer answer than is allowed here. Jewish thought teaches that not only the quality of life is important, but that "human" life itself is sacred and belongs to G-d. We entrust ourselves and our souls to Him. It is an obligation to relieve suffering to the extent that we are able. However, we are not the "masters" of life, even of our own. I would add that allowing the termination of life for any reason opens the possibility of terminating life for lesser reasons. This slippery slope must be avoided.