A matroid $M$ is said to be connected if any two elements of the matroid lie on a common circuit. We say that $S\subseteq E(M)$ is a connected component of $S$ if for all $i,j\in S$ there exists a circuit in $M$ which contains both $i$ and $j$.
I read that if we define a binary relation $R$ on the elements of $M$ such that $i\sim j$ if there exists a circuit in $M$ which contains $i$ and $j$ then $R$ is an equivalence relation and the equivalence classes are the connected components of $M$.
I am having a hard time proving that the relation $R$ is indeed an equivalence relation, in particular the transitivity of the relation. Could anyone give me a hint?
I know that if $i\sim j$ and $j\sim k$ then that there exist two circuits $C_1$ and $C_2$ in $M$ with the first one containing $i,j$ and the second containing $j,k$. Further, the circuit axioms of matroids say that $C_1\cup C_2\setminus \{j\}$ must contain another cycle $C$. I would like to conclude that $i$ and $k$ must both be in $C$ but I don't know why this has to be the case.