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