Gliomas appear with wide variation in their characteristics both in terms of their appearance and location on brain MR images, which makes robust tumour segmentation highly challenging, and leads to high inter-rater variability even in manual segmentations. In this work, we propose a triplanar ensemble network, with an independent tumour core prediction module, for accurate segmentation of these tumours and their sub-regions. On evaluating our method on the MICCAI Brain Tumor Segmentation (BraTS) challenge validation dataset, for tumour sub-regions, we achieved a Dice similarity coefficient of 0.77 for both enhancing tumour (ET) and tumour core (TC). In the case of the whole tumour (WT) region, we achieved a Dice value of 0.89, which is on par with the top-ranking methods from BraTS17-19. Our method achieved an evaluation score that was the equal 5th highest value (with our method ranking in 10th place) in the BraTS20 challenge, with mean Dice values of 0.81, 0.89 and 0.84 on ET, WT and TC regions respectively on the BraTS20 unseen test dataset.