VN-TST 2007 Problem 6

Let $A_{1}A_{2}\ldots A_{9}$ be a regular $9-$ gon. Let $\{A_{1},A_{2},\ldots,A_{9}\}=S_{1}\cup S_{2}\cup S_{3}$ such that $|S_{1}|=|S_{2}|=|S_{3}|=3$ . Prove that there exists $A,B\in S_{1}$ , $C,D\in S_{2}$ , $E,F\in S_{3}$ such that $AB=CD=EF$ and $A \neq B$ , $C\neq D$ , $E\neq F$ .