GeoVLMath: Enhancing Geometry Reasoning in Vision-Language Models via Cross-Modal Reward for Auxiliary Line Creation

Abstract: Auxiliary lines are essential for solving complex geometric problems but remain challenging for large vision-LLMs (LVLMs). Rather than editing diagrams to draw auxiliary lines, which current image editing models struggle to render with geometric precision, we generate textual descriptions of auxiliary-line constructions to better align with the representational strengths of LVLMs. To bridge the gap between textual descriptions and spatial structure, we propose a reinforcement learning framework that enhances diagram-text alignment. At the core of our approach is a cross-modal reward that evaluates how well the generated auxiliary-line description for an original diagram matches a ground-truth auxiliary-line diagram. Built on this reward, we present GeoVLMath, an open-source LVLM tailored to auxiliary-line reasoning in solid geometry. This fine-grained signal drives a GRPO-based RL stage, yielding precise diagram-text alignment. To support training, we develop a scalable data creation pipeline and construct AuxSolidMath, a dataset of 3,018 real-exam geometry problems with paired diagrams and aligned textual fields. At the 3B and 7B scales, GeoVLMath achieves competitive and often superior performance compared with strong open-source and proprietary LVLMs on auxiliary-line reasoning benchmarks.