I think the argument about euthanasia ties in with other issues involving life and death, and so should be answered by taking those into consideration as well.
Those against abortion argue that a fetus is a human being with a right to live. Those in favor of abortion argue that a fetus does not yet have the right to life that all human beings possess.
Those against the death penalty argue that a person, no matter what crime they've committed, retains his or her right to life. Those in favor of the death penalty argue that committing murder constitutes a forfeiture of one's right to life.
I find it interesting that there are many who will argue both against abortion and for the death penalty. Their reasoning is usually that those sentenced to death forfeited their right to life when they committed murder. But my question is this: what gives them the right to decide what constitutes a forfeiture of one's right to live? Should anyone be able to decide this? And if so, whom?
I think that the answer to all three of these issues is rooted in the question of: who should have the power to determine whether or not a person has the right to live? (For the sake of this argument, "person" also includes the unborn and convicted murderers). If you would answer:
no one: Everyone has a right to life which no one, not even the person in question, should have the power to violate. This means that abortion, the death penalty, and euthanasia should all be illegal.
the government/majority: Our leadership should have the power to determine whether someone has the right to live. This meanst hat the legality of abortion, the death penalty, and euthanasia should be decided by the majority opinion and/or the powers that be.
the person in question: Only the person in question should have the power to acknowledge or forfeit his or her right to life. This means that the death penalty should be illegal, euthanasia should be legal, and abortion is still iffy (obviously the fetus doesn't have the capacity to make any such decision, so it could be argued that the right to decide should lie with the person whom the decision most affects - the mother - therefore meaning that abortion should be legal...although one could also argue that since the fetus doesn't have the capacity to decide, and only the person in question can decide, it should be illegal).
I would answer that only the person in question should have the power to forfeit his or her right to life, and therefore I would argue in favor of euthanasia.