Euclid's Elements Book III 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.
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.
Q.E.D.
Proposition 5 <-- Proposition 6 --> Proposition 7