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Euclid: Elementa
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PROPOSITION 5. 
THEOR. 4. PROPOS. 5. 
第五題 
If two circles cut one another, they will not have the same centre. 
SI duo circuli sese mutuò secent; non erit illorum idem centrum. 
兩圜相交。必不同心。 
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. 
   
   
 
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