Find all distinct $a,b,c$ such that $a +1/b = b + 1/c = c+1/a$ Could anyone advise me how to find all values of $a,b, c$ such that $a + \dfrac{1}{b} = b + \dfrac{1}{c} = c + \dfrac{1}{a} \ , $ where $a,b,c $ are distinct real numbers?
I have found that $a, b, c$ can take on either $1,-2, $ or $-\dfrac{1}{2} $, so there are at least 6 sets of solutions. Thank you.

Easy parameterization:
$$
(a,b,c) = \left(a, \frac{1}{1-a}, 1-\frac{1}{a}\right),
$$
where $a$ is any real, $a\ne 0, a\ne \frac{1}{2}, a\ne 1$.
Then $a+\frac{1}{b}=b+\frac{1}{c}=c+\frac{1}{a}=1$.
