Touching Circles Have Different Centers

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Theorem

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

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 $ABC$

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 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.

$\blacksquare$

This is Proposition 6 of Book III of Euclid's The Elements.