Thoth: Theorem-oriented Parsing of Planar Geometry Diagrams with MLLMs

When solving geometry problems, perceiving visual primitives is only the first step; identifying relevant theorems provides essential knowledge for subsequent reasoning. We formulate this process as $\text{what to see}$, then $\text{what knowledge applies}$, and finally $\text{how to deduce}$. Current MLLMs often entangle the latter two within lengthy reasoning trajectories and struggle to identify relevant theorems, leaving subsequent deduction susceptible to unsupported assumptions. We present Thoth, a planar geometry diagram parser that provides both perceived geometric $\text{primitives}$ and question-relevant $\text{theorems}$ to guide subsequent deeper $\text{reasoning}$. To align visual perception with theorem knowledge, we introduce a Theorem-oriented Formal Language (TOFL) and construct GT-210K, a large-scale synthetic dataset with TOFL annotations for scalable supervised fine-tuning. We then perform reinforcement learning with fine-grained process reward signals to further ground theorem predictions in supporting visual evidence and problem-specific constraints. Experiments show that Thoth achieves strong theorem grounding while improving primitive prediction. Across four geometry VQA benchmarks, the grounded primitives and theorems effectively improve downstream reasoning performance, demonstrating their value as explicit knowledge for subsequent deduction. Code, datasets, and models will be publicly available upon acceptance.