Automatically constructing taxonomy finds many applications in e-commerce and web search. One critical challenge is as data and business scope grow in real applications, new concepts are emerging and needed to be added to the existing taxonomy. Previous approaches focus on the taxonomy expansion, i.e. finding an appropriate hypernym concept from the taxonomy for a new query concept. In this paper, we formulate a new task, “taxonomy completion”, by discovering both the hypernym and hyponym concepts for a query. We propose Triplet Matching Network (TMN), to find the appropriate pairs for a given query concept. TMN consists of one primal scorer and multiple auxiliary scorers. These auxiliary scorers capture various fine-grained signals (e.g., query to hypernym or query to hyponym semantics), and the primal scorer makes a holistic prediction on triplet based on the internal feature representations of all auxiliary scorers. Also, an innovative channel-wise gating mechanism that retains task-specific information in concept representations is introduced to further boost model performance. Experiments on four real-world large-scale datasets show that TMN achieves the best performance on both taxonomy completion task and the previous taxonomy expansion task, outperforming existing methods.