MOTIVATION: The drug-disease, gene-disease, and drug-gene relationships, as high-frequency edge types, describe complex biological processes within the biomedical knowledge graph. The structural patterns formed by these three edges are the graph motifs of (disease, drug, gene) triplets. Among them, the triangle is a steady and important motif structure in the network, and other various motifs different from the triangle also indicate rich semantic relationships. However, existing methods only focus on the triangle representation learning for classification, and fail to further discriminate various motifs of triplets. A comprehensive method is needed to predict the various motifs within triplets, which will uncover new pharmacological mechanisms and improve our understanding of disease-gene-drug interactions. Identifying complex motif structures within triplets can also help us to study the structural properties of triangles. RESULTS: We consider the seven typical motifs within the triplets and propose a novel graph contrastive learning-based method for triplet motif prediction (TriMoGCL). TriMoGCL utilizes a graph convolutional encoder to extract node features from the global network topology. Next, node pooling and edge pooling extract context information as the triplet features from global and local views. To avoid the redundant context information and motif imbalance problem caused by dense edges, we use node and class-prototype contrastive learning to denoise triplet features and enhance discrimination between motifs. The experiments on two different-scale knowledge graphs demonstrate the effectiveness and reliability of TriMoGCL in identifying various motif types. In addition, our model reveals new pharmacological mechanisms, providing a comprehensive analysis of triplet motifs. AVAILABILITY AND IMPLEMENTATION: Codes and datasets are available at https://github.com/zhanglabNKU/TriMoGCL and https://doi.org/10.5281/zenodo.14633572.