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.