A circle does not touch another circle at more than one point whether it touches it internally or externally.

For, if possible, let the circle *ABDC* touch the circle *EBFD,* first internally, at more points than one, namely *D* and *B.*

Take the center *G* of the circle *ABDC* and the center *H* of *EBFD.*

Therefore the straight line joined from *G* to *H* falls on *B* and *D.*

Let it so fall, as *BGHD.*

Then, since the point *G* is the center of the circle *ABCD* and *BG* equals *GD,* therefore *BG* is greater than *HD.* Therefore *BH* is much greater than HD.

Again, since the point *H* is the center of the circle *EBFD, BH* equals *HD,* but it was also proved much greater than it, which is impossible.

Therefore a circle does not touch a circle internally at more points than one.

I say further that neither does it so touch it externally.

For, if possible, let the circle *ACK* touch the circle *ABDC* at more points than one, namely *A* and *C.* Join *AC.*

Then, since on the circumference of each of the circles *ABDC* and *ACK* two points *A* and *C* have been taken at random, the straight line joining the points falls within each circle, but it fell within the circle *ABCD* and outside *ACK,* which is absurd.

Therefore a circle does not touch a circle externally at more points than one.

And it was proved that neither does it so touch it internally.

Therefore *a circle does not touch another circle at more than one point whether it touches it internally or externally.*

Q.E.D.

There are logical flaws in this proof similar to those in the last two proofs.

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