If two circles cut one another, they will not have the same centre.
For let the circles ABC, CDG cut one another at the points B, C; I say that
they will not have the same centre.
For, if possible, let it be E; let EC be joined, and let EFG be drawn through
at random.
Then, since the point E is the centre of the circle ABC, EC is equal to EF
[I. Def. 15].
Again, since the point E is the centre of the circle CDG, EC is equal to EG.
But EC was proved equal to EF also;
therefore EF is also equal to EG, the less to the greater: which is impossible.
Therefore the point E is not the centre of the circles ABC, CDG.
Therefore, etc.
Q.E.D.