Intersecting Circles have Different Centers

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In the words of Euclid:

If two circles cut one another, they will not have the same centre.

(The Elements: Book $\text{III}$: Proposition $5$)



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 $\text{I}$ 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.


Historical Note

This proof is Proposition $5$ of Book $\text{III}$ of Euclid's The Elements.