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

arXiv: https://arxiv.org/abs/2510.11020 · HTML: https://mlsanity.com/paper/2510.11020/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Placed in Spatial and embodied reasoning, one of 17 directions in LLM reasoning.
- Builds on 2 of its 38 references that are in the map.
- First posted on arXiv 13 Oct 2025.

## Placed in

- LLM reasoning / [Spatial and embodied reasoning](https://mlsanity.com/lr/spatial-and-embodied-reasoning/)
- Multimodal LLMs / vision-language models / [Embodied and 3D vision-language models](https://mlsanity.com/mllm/embodied-and-3d-vision-language-models/)
- LLM alignment / RLHF / preference optimisation / [Diffusion and multimodal model alignment](https://mlsanity.com/alignment/diffusion-and-multimodal-model-alignment/)

## Builds on (most cited first, 2 of 2)

- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [GeoLaux: A Benchmark for Evaluating MLLMs' Geometry Performance on Long-Step Problems Requiring Auxiliary Lines](https://mlsanity.com/paper/2508.06226/) (2025)

## Built on by (most cited first, 8 of 8)

- [Skyra: AI-Generated Video Detection via Grounded Artifact Reasoning](https://mlsanity.com/paper/2512.15693/) (2025)
- [AdaptMMBench: Benchmarking Adaptive Multimodal Reasoning for Mode Selection and Reasoning Process](https://mlsanity.com/paper/2602.02676/) (2026)
- [DIVA-GRPO: Enhancing Multimodal Reasoning through Difficulty-Adaptive Variant Advantage](https://mlsanity.com/paper/2603.01106/) (2026)
- [DynaSolidGeo: A Dynamic Benchmark for Genuine Spatial Mathematical Reasoning of VLMs in Solid Geometry](https://mlsanity.com/paper/2510.22340/) (2025)
- [Geo-Code: A Code Framework for Reverse Code Generation from Geometric Images Based on Two-Stage Multi-Agent Evolution](https://mlsanity.com/paper/2602.07749/) (2026)
- [Geoparsing: Diagram Parsing for Plane and Solid Geometry with a Unified Formal Language](https://mlsanity.com/paper/2604.11600/) (2026)
- [PAGER: Bridging the Semantic-Execution Gap in Point-Precise Geometric GUI Control](https://mlsanity.com/paper/2605.15963/) (2026)
- [Draw2Think: Harnessing Geometry Reasoning through Constraint Engine Interaction](https://mlsanity.com/paper/2605.20743/) (2026)

## Abstract (arXiv)

Auxiliary lines are essential for solving complex geometric problems but remain challenging for large vision-language models (LVLMs). Recent attempts construct auxiliary lines via code-driven rendering, a strategy that relies on accurate and executable code generation to produce visual renderings of the auxiliary lines for subsequent reasoning. However, in complex solid geometry settings, such a strong dependence on precise specifications substantially restricts the robustness of this strategy. Alternatively, we turn to a simpler and more stable solution, representing auxiliary-line constructions as structured textual descriptions. To bridge the gap between textual descriptions and spatial structure, we propose a reinforcement learning framework that enhances diagram-text alignment. The core is a cross-modal reward model that evaluates how well the generated auxiliary-line description matches the ground-truth auxiliary-line diagram. The reward signal drives a GRPO-based RL stage to yield informative auxiliary-line descriptions for the reasoning. To support the training and evaluation, we develop a scalable data pipeline and construct AuxSolidMath, a dataset of 3,018 real-exam geometry problems with paired diagrams and aligned textual fields. Based on this framework, we derive GeoVLMath, an LVLM for solving complex solid geometry.