Touching Circles have Different Centers

Theorem

In the words of Euclid:

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

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.

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.