Euclid's Elements, Book III, Proposition 6

Proposition 6

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.		I.Def.15
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.

As mentioned before, this proposition is almost the same as the previous. Both could be included in one statement: circles that meet don't have the same center, or the contrapositive: concentric circles don't meet.

This propostion is not used in the rest of the Elements.

Book III Introduction - Proposition III.5 - Proposition III.7.