Find all $a,b,c\in\mathbb{Z}_{\neq0}$ with $\frac ab+\frac bc=\frac ca$
Write $x=a/b$,$y=b/c$, $z=c/a$ so get the system $x+y=z$ and $x y z=1$, therefore $x y (x+y)=1$, with $x$, $y$,$z$ rationals. Now write $x=m/q$, $y=n/q$ and we get $$m/q \cdot n/q \cdot (m+n)/q = 1$$ or $$m n (m+n) = q^3$$
We may assume that the numbers $m$, $n$, $m+n$ are pairwise relatively prime, otherwise get rid of the common factor. Therefore, $m$, $n$ and $m+n$ are all cubes, impossible ( Fermat for cubes, proved by Euler).