Aseres Hadibros: Lo Sirtzach - Who Has the Right To End Life?
When someone is killed it diminishes the presence of Hashem in this world. Man was made in the image of Hashem. Even if a person has given up on his life and begs you to kill him, it is not permissible to kill him. Each person has a G-dly neshamsa, and it is not only forbidden to kill others but it is also forbidden to commit suicide. You may not forfeit your own life because you do not have the right to take away the image and presence of Hashem in this world.