# Touching Circles have Different Centers

## 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.

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 proof is Proposition $6$ 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