Case 1: If $a=b$, then $a=b=c$ and we are done. Case 2: If $a\ne b$, then we can let $|a-b|=6$ WLOG. Then $|b-c|=3$ and $|c-a|=2$. Then triangle inequality states that $|a-b|\le |b-c|+|c-a|$. However, $6>3+2$ so this case is impossible. Therefore, $a=b=c$.