Human Euthanasia

Recently on the Fugly blog, they have been discussing humane euthanasia in the case of "rescued" horses.  It got me thinking about another type of euthanasia, one which is probably more controversial than whether or not to put down a horse.

I'm talking about human euthanasia, specifically in people who will never recover from things like Alzheimer's, or who would already be dead if a machine weren't breathing for them.  Generally, I think people should be allowed to die when it is clear their quality of life is suffering a great deal.  I am trying to be as broad as possible in this, as there are always exceptions to the rule.

I have one grandma who has entered the advanced stages of Alzheimer's, and occasionally has refused to eat.  Eventually, the nurses get her eating again but she isn't the woman who used to be my grandmother.  My grandmother would never have allowed herself to get to the state she is in, but Alzheimer's is a sneaky thief who steals your memories, thoughts, and wishes from you before you know it.  I love my grandmother deeply, and that is why I believe she should be allowed to die.  It seems like the only humane thing to do- I would put my dog to sleep if he ever got as far gone as she is.  She struggles to remember anyone, even her husband and six children.

So what do you think?  Is this too slippery a slope? I know there are terrible people who would "let go" of their "loved ones" just to get an inheritance, or stop paying medical bills, or because Mom always liked the other sibling better.  What is the right course of action?