Intersecting Circles Have Different Centers

Theorem

If two circles cut one another, then they do not have the same center.

Geometric Proof

Let $ABC$ and $BDCG$ be circles which cut one another at $B$ and $C$.

Suppose they had the same center $E$.

Join $EC$ and let $EG$ be drawn at random through $F$.

As $E$ is the center of $ABC$, by Book I Definition 15: Circle, we have that $EC = EF$.

Similarly, as $E$ is also the center of $BDCG$, we have that $EC = EG$.

But they are clearly unequal by the method of construction.

So from this contradiction, the two circles can not have the same center.

$\blacksquare$

Historical Note

This is Proposition 5 of Book III of Euclid's The Elements.