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