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Recent work on curvilinear structure segmentation has mostly focused on backbone network design and loss engineering. The challenge of collecting labelled data, an expensive and labor intensive process, has been overlooked. While labelled data is expensive to obtain, unlabelled data is often readily available. In this work, we propose SemiCurv, a semi-supervised learning (SSL) framework for curvilinear structure segmentation that is able to utilize such unlabelled data to reduce the labelling burden. Our framework addresses two key challenges in formulating curvilinear segmentation in a semi-supervised manner. First, to fully exploit the power of consistency based SSL, we introduce a geometric transformation as strong data augmentation and then align segmentation predictions via a differentiable inverse transformation to enable the computation of pixel-wise consistency. Second, the traditional mean square error (MSE) on unlabelled data is prone to collapsed predictions and this issue exacerbates with severe class imbalance (significantly more background pixels). We propose a N-pair consistency loss to avoid trivial predictions on unlabelled data. We evaluate SemiCurv on six curvilinear segmentation datasets, and find that with no more than 5% of the labelled data, it achieves close to 95% of the performance relative to its fully supervised counterpart.