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