Mercy killing is a negation of all that we believe in. Mercy killing negates the belief that Allah is active in running the affairs of the individual who is being subjected to euthanasia. Mercy killing negates our belief in Allah's absolute mercy and absolute wisdom. Mercy killing negates our contentment in Allah's plans. Mercy killing negates our belief in the fact that the life of this world, from birth till death is a test for us.
Mercy killing is against steadfastness.
Thus, if euthanasia means to stop sustaining life that can only be sustained through artificial means, then there is nothing wrong with it. If a person is put to sleep to avoid the pain and suffering that he is likely to suffer if he is allowed to live, then this would be taking life. In my opinion, this cannot be allowed in an Islamic society. The time of a persons death and the amount of the suffering that he must go through is for Allah, the most wise and the ever merciful to decide. Man, with the knowledge that he has gained in the field of medicine, must try to find cures to all the diseases and also try to make a death as painless as possible.