Intersecting Circles have Different Centers

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

Geometric Proof

 * Euclid-III-5.png

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, 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.