If two circles touch one another, they will not have the same centre.
For let the two circles ABC, CDE touch one another at the point C; I say that
they will not have the same centre.
For, if possible, let it be F; let FC be joined, and let FEB be drawn through
at random.
Then, since the point F is the centre of the circle ABC, FC is equal to FB.
Again, since the point F is the centre of the circle CDE, FC is equal to FE.
But FC was proved equal to FB; therefore FE is also equal to FB, the less to
the greater: which is impossible.
Therefore F is not the centre of the circles ABC, CDE.
Therefore, etc.
Q.E.D.