It sounds to me like the original question may involve a debate in which they have to take the side that euthanasia is inherently wrong. While I disagree with that position, I will try to help you out.
The various points raised above can be used as arguments for not allowing any euthanasia -- they are so fuzzy and difficult that it is better to avoid them and err on the side of protecting human life. There are many, many of these sorts of issues.
Unless there is a crystal clear living will, how do we know that the decision makers aren't following their own desires ($$, getting rid of a burden, etc.) rather than the best interest of the patient?
If the person in question is asking for euthanasia, how do we decide if they are truly competent to decide rather than under the influence of depression?
Has God declared it an unforgiveable sin to take human life under any circumstances?
Is enduring suffering an important part of life and the human condition? Perhaps an important passage that the soul needs to go through?
Are the doctor or family members politically motivated by a desire to advocate for the right to die?
There is one distinction that is really important!!! Euthanasia is NOT the same as not taking heroic measures to keep someone alive. Euthanasia involves helping someone with affirmative steps to end their life. The first is a failure to keep them alive -- you aren't killing them, but are letting them die if that is the natural course. The second -- euthanasia -- goes much farther. It causes death.
I am in complete accord with accessteacher's caution on conflict of interest if a family member who stands to receive an inheritance is permitted to make a euthanasia decision. There are multiple troubling ethical concerns surrounding this issue.
Nonetheless, I have watched several elderly family members go through an agonizing loss of dignity, not to mention significant physical pain, as they lingered on in a hospital bed. I now have a living will and a health care proxy to prevent this from ever happening to me. If I were to reach such a state, I'd rather my family make the decision to "pull the plug" than to just live on in a vegetative state. If I were to discover that I have a degenerative disease such as ALS or Alzheimer's, I'd rather be in the position to make my own choice as to when I would exit with dignity than to deteriorate in front of my family.
In addition to the comments above, one of the critical questions that needs to be discussed and resolved is the influence of the family in making the decision. You can easily see a conflict of interests if you are deciding whether a family member should be allowed to "die well" if you are going to receive a lot of money, especially if the alternative is for the family member to receive hospice care that will eat in to your inheritance. Therefore whilst dying with dignity is something we can all agree with, it does raise a whole number of other issues that need to be carefully considered.
Euthanasia does not strike me as wrong in the idea of putting someone who is suffering out of his or her misery. I am thinking of an obviously terminal illness such as cancer. I am for living wills and the avoidance of "heroic" medical procedures to prolong a life that is going to end in a very short time with great pain and suffering. I am for the administration of very strong pain killers to ease the suffering of terminally ill persons.
I believe it is unethical to administer medications to end human life. The difficulty in this concept is that someone must decide who lives and who dies. What is a life worth living? Who would ultimately decide these things?
My opinion is that euthanasia per se is not wrong. However, it involves the taking of a human life and therefore raises profound moral, ethical, and legal questions. Pro-life people would argue that the taking of any human life for any reason is immoral and unethical. They usually offer a religious underpinning for their reasoning.
I believe that in extremely limited cases, euthanasia can be a blessing to a suffering individual. The problem comes in trying to establish exactly what is an acceptable circumstance. Moreover, once euthanasia is deemed acceptable, who will decide what circumstances will limit the practice, if any? These are the basic issues that surround the dilemma of whether euthanasia is acceptable.
Euthanasia can have two types of negative consequences - for the individual who loses life, and the society in general. Practice of euthanasia can impact the values and culture of a society negatively, leading to long term reduction on value placed by the people on value of life.
While it may be possible to justify euthanasia under certain conditions it is not easy to to define these conditions - that is, the state of health when euthanasia may be considered and the method of giving effect to it.
I believe positive action to hasten end of life is deplorable under all conditions, whereas withdrawal of treatment to terminally ill patients under some conditions may be justified at the request of person concerned.