Euthanasia is to put to death someone who is suffering - even if the person being killed is in agreement with the act. Do you know the Church's position on it?
"A person who participates in euthanasia, including so-called assisted suicide, violates the commandments of God" (CHI)
It seems fairly clear and unambiguous.
However, (you just knew I had a 'however' in there, right?)....
If my wife or children were facing, not a medical problem, but torture by some of the barbarians in this world, I might have to reconsider such a thing. A quick, painless death, IMO would be far preferable to them having to endure weeks, months, or years of torment.
For medical problems, I think that there is probably always some alternative, whether it is in better pain medication, or prayer and the ordinances of the Priesthood.
A final 'however' does apply. In the case of a person who is completely dependant on machines to keep them alive. For myself, I would probably want to have the machines turned off, and let the natural processes occur. In this case, although it really is considered euthanasia, I wouldn't agree. It is not actively causing the death, it is allowing it to happen. I wouldn't agree to stopping food and water, even if they require unusual means to provide them. But if a person is unable to eat, drink, breathe, on their own, without machines, then I think it is valid to allow the process to complete.
As a final note. After I donated a kidney, I found out that a man in my brother's ward had had three kidney transplants. When the third began to fail, he refused any further possibility of another transplant, or even dialysis. He felt that his body was completely worn out, and that his life had been artificially extended long enough. He was fairly young, in his 40's or 50's.
In this case, I would probably have agreed with him. I know that a lot of people were really upset with him, considering that he was committing suicide. But he wasn't really. He was letting the natural process occur, without any attempt to stop it any more. I really hope I am not wrong in this, as I would take the same attitude.
Nighthawk, although that is different to the this topic... I think it is very interesting the part you said about a quick death to save long term suffering of family members from barbaric acts I will start a thread on that... let us see what the others say.
Anyway, as far as euthanasia goes the Church they just see it as cut and dry.