Dear Joseph:

Well the first rule of apologetics is that it is not our job to convince anyone of our Faith. Our job is to be the messenger, to present the faith to people in as convincing and persuasive way as we can, but to leave the convincing to the Holy Spirit. Jesus said that,  "No one can come to me unless the Father who sent me draws him..." (John 6:44).

What you must do is to simply explain the Catholic position on Euthanasia. Euthanasia is deliberately taking someone's life. That is the very definition of murder. It is a violation of the the very essence and dignity of Life.

Dignity is not found in killing someone because they are sick or infirmed or in killing oneself. The Doctrine of Suffering is very much a part of this (see the Q&A I just answered about suffering). Suffering is redemptive. It is a privilege to join our suffering with that of Christ. Atheists, and similarly thinking people, will have no understanding of this. All we can do is to appeal to the truth and let the Holy Spirit work on their conscience.

This appeal can begin explanation of Church teaching in the Catechism:

Euthanasia

2276 Those whose lives are diminished or weakened deserve special respect. Sick or handicapped persons should be helped to lead lives as normal as possible.

2277 Whatever its motives and means, direct euthanasia consists in putting an end to the lives of handicapped, sick, or dying persons. It is morally unacceptable.

Thus an act or omission which, of itself or by intention, causes death in order to eliminate suffering constitutes a murder gravely contrary to the dignity of the human person and to the respect due to the living God, his Creator. The error of judgment into which one can fall in good faith does not change the nature of this murderous act, which must always be forbidden and excluded.

2278 Discontinuing medical procedures that are burdensome, dangerous, extraordinary, or disproportionate to the expected outcome can be legitimate; it is the refusal of "over-zealous" treatment. Here one does not will to cause death; one's inability to impede it is merely accepted. The decisions should be made by the patient if he is competent and able or, if not, by those legally entitled to act for the patient, whose reasonable will and legitimate interests must always be respected.

2279 Even if death is thought imminent, the ordinary care owed to a sick person cannot be legitimately interrupted. The use of painkillers to alleviate the sufferings of the dying, even at the risk of shortening their days, can be morally in conformity with human dignity if death is not willed as either an end or a means, but only foreseen and tolerated as inevitable Palliative care is a special form of disinterested charity. As such it should be encouraged.

I also suggest that you read the article, What About the Right to Die? Here's how to answer the common arguments of the culture of death by Fr. Frank Pavone.

With this information you can answer the basic questions about Church teaching on euthanasia. But, arguing the case is not productive. Avoid the bait and trap of debate. Merely state what the Church teaches. It your opponent does not like those answers, that is his problem.

In the end, I would say, "Look, there is nothing to argue here. This is what the Church teaches and what the Church teaches is what God teaches about the dignity and sanctity of life. You can accept it or reject it, that is your choice, and you will be held accountable for your choice, as we all will be."

At that point, the opponent will often try to bait you into argument about the Church killing people in the Crusades or the Inquisition. Since such an opponent is not interested in the truth, I would not debate the person. Rather, I would reply, "Your assertions about the Inquisition are not true, which can be proven with documentary evidence. But, since you are merely trying to bait me and are not interested in the truth, I shall not indulge you. Good day."

Most likely the person will then call you names and accuse you of hiding the truth about the Church, or goad you with, "you can't prove your position can you", or "you are a liar", or "you just don't want to hear the truth" or "you know I am right about this" or "you and your Church are cruel and heartless", or a whole host of other nonsense. Resist the temptation to reply. Allow the person to think his stupid thoughts and opinions. Do not give in to what is actually the pride of response. Just walk away and let the person blather away. Let him stew in the juices of his own stupidity.

St. Paul said it well in teaching us the principle of avoiding unproductive argumentation:

But avoid stupid controversies, genealogies, dissensions, and quarrels over the law, for they are unprofitable and futile. As for a man who is factious [divisive, heretical], after admonishing him once or twice, have nothing more to do with him, knowing that such a person is perverted and sinful; he is self-condemned. (Titus 3:9-11)

Then pray for that person that he will someday be open to the Holy Spirit to come to a knowledge and faith in the Truth.

God Bless,
Bro. Ignatius Mary