I have no problem with euthanasia, per se, but there need to be a lot of checks and balances. For adults, of course, simple consent is all that should be needed. As thir stated, you own your own life. If you are unable to make an informed, rational decision due to illness or loss of mental capacity, your next of kin should be allowed to make that decision. A living will, naming the person able to make such decisions, should be everyone's priority. For the record, this was my mother's condition, her mind destroyed by Alzheimer's, her body failing all too slowly. My dad had the legal right to choose for her. And he did. And I supported him in it.
For children, though, we have other problems. Yes, a child over the age of about 12 is probably capable of understanding what such a decision means, and should be able to make that decision with his parents; and doctors, consent. Younger children (and I don't claim to know just how young this should be) may not truly understand the implications, but for sure their parents should. And their doctors definitely should. So a parent, knowing that the doctors can do nothing more, and that their child is in pain with no hope in sight, should be allowed to make that choice.
One thing I do know, however: they need to keep the fucking churches out of the loop! If the parents, or patient, are religious, then it is up to them whether or not to abide by the strictures of their church. If they are not religious, or reject those strictures, the churches should not be permitted to force them into not going forward. They can organize all the prayer vigils they like, since those will have no effect on reality anyway. And they can preach to their own people as much as they like, too. Those who rely on the authority of the church to make their decisions for them will likely follow suit. But the churches should not be allowed to make such decisions for the public at large.