Let $\mathcal{A}$ be an additive category. And let $C(\mathcal{A})$ be the category of the chain complexes. Now let $f:X_* \rightarrow Y_*$ be a chain map in $C(\mathcal{A})$ and $C(f)$ be its mapping cone. As we've already known, there's a proposition:
If $C(f)$ is contractible, then $f:X_* \rightarrow Y_*$ is a homotopy equivalence.
My question: Is the inverse problem true? To be explicitly, if $f:X_* \rightarrow Y_*$ is a homotopy equivalence, then does $C(f)$ necessarily to be contractible? I guess that this is not true in general for a part of the proof of the proposition doesn't holds conversely. Am I right? If I'm right, is there any counterexample? Thanks in advance.