Touching Circles have Different Centers

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Theorem

In the words of Euclid:

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

(The Elements: Book $\text{III}$: Proposition $6$)


Proof

If the two circles are outside one another, the result is trivial.

This proof will focus on the situation where one circle is inside the other one.


Euclid-III-6.png

Let $ABC$ and $CDE$ be circles which touch one another at $C$, such that $CDE$ is inside $ABC$


Aiming for a contradiction, suppose they had the same center $F$.

Join $FC$ and let $FB$ be drawn at random through $E$.

As $F$ is the center of $ABC$, by Book $\text{I}$ Definition $15$: Circle, we have that $FB = FC$.

Similarly, as $F$ is also the center of $CDE$, we have that $FC = FE$.

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 theorem is Proposition $6$ of Book $\text{III}$ of Euclid's The Elements.


Sources