If two circles touch one another, then they do not have the same center.

Let the two circles *ABC* and *CDE* touch one another at the point *C.*

I say that they do not have the same center.

For, if possible, let it be *F.* Join *FC,* and draw *FEB* through at random.

Then, since the point *F* is the center of the circle *ABC, FC* equals *FB.* Again, since the point *F* is the center of the circle *CDE, FC* equals *FE.*

But *FC* was proved equal to *FB,* therefore *FE* also equals *FB,* the less equals the greater, which is impossible.

Therefore *F* is not the center of the circles *ABC* and *CDE.*

Therefore *if two circles touch one another, then they do not have the same center.*

Q.E.D.

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