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.
∎
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"