Intersecting Circles have Different Centers
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Theorem
In the words of Euclid:
(The Elements: Book $\text{III}$: Proposition $5$)
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 $\text{I}$ 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.
$\blacksquare$
Historical Note
This proof is Proposition $5$ of Book $\text{III}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{III}$. Propositions