For a vector $w$, let $T\_{w}$ be the translation by $w$.


I was told that the following observation about subsets of the plane was due to H. Hopf:


Let $X$ be a compact, path-connected subset of the plane. Then, if for some vector $v$, $X\cap T\_{v}[X]\neq \emptyset$, then for each positive natural number $n$, $X\cap T\_{\frac{1}{n} \cdot v} \neq \emptyset$.


I have a proof (which may or may not be Hopf's), but what I want is the reference to cite. 


My understanding is that Hopf used is to prove that if a 1-sphere properly embedded on the torus represents the homology element $al+bm$ (where $l$ and $m$ are the longitude and meridian) then $a$ and $b$ are relatively prime.



Hopf reference sought
Check out the Chord Theorem (Theorem 2B12) in Rolfsen's Knots and Links. Here is a link to the [Google book](http://books.google.com/books?id=s4eGEecSgHYC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false).


