Suppose $R$ is a transitive binary relation that contains a cycle $a_1Ra_2$, $a_2Ra_3$, $\dots$, $a_{n-1}Ra_n$, $a_nRa_1$. Does this imply that $R$ is symmetric for any pairs in this cycle, i.e. $a_iRa_j\Rightarrow a_jRa_i$ for all $a_i, a_j\in \{a_1, \dots, a_n\}$?
If so, how do I prove it?
Let us consider induction on $n$ of a slightly stronger statement. If $R$ is a transitive binary relation that contains a cycle of length $n$, then $xRy$ for any $x$ and $y$ in the cycle. In this case we will say $R$ is full with respect to the cycle.
Base case $n=1$ and $n=2$ are trivial.
Inductive step. Assume that whenever there is a cycle of length $n\geq 2$ then the relation is full with respect to the cycle.. Now let $(a_1Ra_2, a_2Ra_3, ..., a_nRa_{n+1}, a_{n+1}Ra_1)$ be a cycle of length $n+1\geq 3$ and let $a_i$ and $a_j$ be any two elements of the cycle. Since $n+1\geq 3$ we may assume WLOG that $2\neq i$ and $2\neq j$. Then since the relation is transitive $a_1Ra_2$ and $a_2Ra_3$ implies $a_1Ra_3$ so we have a cycle $( a_1Ra_3, ..., a_nRa_{n+1},a_{n+1}Ra_1)$ of length $n$ that contains $a_i$ and $a_j$. Thus $a_iRa_j$.
Thus by induction we have that If $R$ is a transitive binary relation that contains a cycle, then $R$ is full with respect to the cycle.
Then symmetry follows trivially from that.