Proof: By Euclid
(related to Proposition: 3.06: Touching Circles have Different Centers)
- For, if possible, let $F$ be (the common center), and let $FC$ have been joined, and let $FEB$ have been drawn through (the two circles), at random.
- Therefore, since point $F$ is the center of the circle $ABC$, $FC$ is equal to $FB$.
- Again, since point $F$ is the center of the circle $CDE$, $FC$ is equal to $FE$.
- But $FC$ was shown (to be) equal to $FB$.
- Thus, $FE$ is also equal to $FB$, the lesser to the greater.
- The very thing is impossible.
- Thus, point $F$ is not the (common) center of the circles $ABC$ and $CDE$.
- Thus, if two circles touch one another then they will not have the same center.
- (Which is) the very thing it was required to show.
Thank you to the contributors under CC BY-SA 4.0!
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"