It seems like the majority of Christian denominations oppose euthanasia, although I can't imagine it's universal.
What is the biblical basis against euthanizing terminally ill and those who are diagnosed as brain dead?
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The biblical argument, I believe, is that as Christians we all must have faith, and just as Christ was able to wake the "dead girl who was sleeping" (Mat. 9) so we should have faith that Christ can heal. Things get a little bit more ambiguous when you're talking about brain death, though — we are able to keep body parts alive long after the original owner has clearly passed on (a transplanted heart will often out-survive its original owner for years), but committing suicide, in any circumstance, has been considered a profound lack of hope.
A non-biblical argument (put forth by Augustine) was that, more or less, "dying" had no substantive meaning — I could have cancer and be told that i have two weeks to live. Clearly I would be "dying" in that case, but I could then make a full recovery tomorrow. On the other hand, I could be perfectly healthy and then decide to go for a walk and get hit by a bus. In this case, the person who seemed healthy was the one who was really dying and the person who seemed sick was not dying at all!
This may be overly simplistic, but I think it gets straight to the point:
Thou shalt not kill. - Exodus 20:13 (King James Version)