If in a circle two straight lines which do not pass through the center cut one another, then they do not bisect one another.

Let *ABCD* be a circle, and in it let the two straight lines *AC* and *BD,* which do not pass through the center, cut one another at *E.*

I say that they do not bisect one another.

For, if so, let them bisect one another, so that *AE* equals *EC,* and *BE* equals *ED.* Take the center *F* of the circle *ABCD.* Join *FE.*

Then, since a straight line *FE* passing through the center bisects a straight line *AC* not passing through the center, it also cuts it at right angles, therefore the angle *FEA* is right.

Again, since a straight line *FE* bisects a straight line *BD,* it also cuts it at right angles. Therefore the angle *FEB* is right.

But the angle *FEA* was also proved right, therefore the angle *FEA* equals the angle *FEB,* the less equals the greater, which is impossible.

Therefore *AC* and *BD* do not bisect one another.

Therefore *if in a circle two straight lines which do not pass through the center cut one another, then they do not bisect one another.*

Q.E.D.

This proposition is not used in the rest of the *Elements.*