I say that they do not bisect one another.
For, if possible, let them bisect one another, so that AE is equal to EC, and BE to ED; let the centre of the circle ABCD be taken [[III. 1](elem.3.1)], and let it be F; let FE be joined.

Then, since a straight line FE through the centre bisects a straight line AC not through the centre, it also cuts it at right angles; [[III. 3](elem.3.3)] therefore the angle FEA is right.

Again, since a straight line FE bisects a straight line BD, it also cuts it at right angles; [[III. 3](elem.3.3)] therefore the angle FEB is right.

But the angle FEA was also proved right; therefore the angle FEA is equal to the angle FEB, the less to the greater: which is impossible.

Therefore AC, BD do not bisect one another.

Therefore etc. Q. E. D.