# GeoFM: Enhancing Geometric Reasoning of MLLMs via Synthetic Data Generation through Formal Language

arXiv: https://arxiv.org/abs/2510.27448 · HTML: https://mlsanity.com/paper/2510.27448/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Spatial and embodied reasoning, one of 17 directions in LLM reasoning.
- Builds on 23 of its 43 references that are in the map.
- First posted on arXiv 31 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/)
- Synthetic data and self-improvement for LLMs / [Self-improving MLLMs and synthetic data](https://mlsanity.com/synth/self-improving-mllms-and-synthetic-data/)

## Builds on (most cited first, 23 of 23)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [GPT-4o System Card](https://mlsanity.com/paper/2410.21276/) (2024)
- [Qwen2-VL: Enhancing Vision-Language Model's Perception of the World at Any Resolution](https://mlsanity.com/paper/2409.12191/) (2024)
- [Qwen-VL: A Versatile Vision-Language Model for Understanding, Localization, Text Reading, and Beyond](https://mlsanity.com/paper/2308.12966/) (2023)
- [MathVista: Evaluating Mathematical Reasoning of Foundation Models in Visual Contexts](https://mlsanity.com/paper/2310.02255/) (2023)
- [ChatGLM: A Family of Large Language Models from GLM-130B to GLM-4 All Tools](https://mlsanity.com/paper/2406.12793/) (2024)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Gemini: A Family of Highly Capable Multimodal Models](https://mlsanity.com/paper/2312.11805/) (2023)
- [WizardMath: Empowering Mathematical Reasoning for Large Language Models via Reinforced Evol-Instruct](https://mlsanity.com/paper/2308.09583/) (2023)
- [Measuring Multimodal Mathematical Reasoning with MATH-Vision Dataset](https://mlsanity.com/paper/2402.14804/) (2024)
- [We-Math: Does Your Large Multimodal Model Achieve Human-like Mathematical Reasoning?](https://mlsanity.com/paper/2407.01284/) (2024)
- [Enhancing the Reasoning Ability of Multimodal Large Language Models via Mixed Preference Optimization](https://mlsanity.com/paper/2411.10442/) (2024)
- [G-LLaVA: Solving Geometric Problem with Multi-Modal Large Language Model](https://mlsanity.com/paper/2312.11370/) (2023)
- [MAVIS: Mathematical Visual Instruction Tuning with an Automatic Data Engine](https://mlsanity.com/paper/2407.08739/) (2024)
- [GeomVerse: A Systematic Evaluation of Large Models for Geometric Reasoning](https://mlsanity.com/paper/2312.12241/) (2023)
- [Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2](https://mlsanity.com/paper/2502.03544/) (2025)
- [PP-OCRv2: Bag of Tricks for Ultra Lightweight OCR System](https://mlsanity.com/paper/2109.03144/) (2021)
- [FormalGeo: An Extensible Formalized Framework for Olympiad Geometric Problem Solving](https://mlsanity.com/paper/2310.18021/) (2023)
- [EAGLE: Elevating Geometric Reasoning through LLM-empowered Visual Instruction Tuning](https://mlsanity.com/paper/2408.11397/) (2024)
- [Automatically Building Diagrams for Olympiad Geometry Problems](https://mlsanity.com/paper/2012.02590/) (2020)

## Built on by (most cited first, 3 of 3)

- [PAGER: Bridging the Semantic-Execution Gap in Point-Precise Geometric GUI Control](https://mlsanity.com/paper/2605.15963/) (2026)
- [GeoSym127K: Scalable Symbolically-verifiable Synthesis for Multimodal Geometric Reasoning](https://mlsanity.com/paper/2605.16371/) (2026)
- [Reactivating Test-Time Scaling for Plane Geometry Problem Solving](https://mlsanity.com/paper/2608.30156/) (2026)

## Abstract (arXiv)

Multi-modal Large Language Models (MLLMs) have gained significant attention in both academia and industry for their capabilities in handling multi-modal tasks. However, these models face challenges in mathematical geometric reasoning due to the scarcity of high-quality geometric data. To address this issue, synthetic geometric data has become an essential strategy. Current methods for generating synthetic geometric data involve rephrasing or expanding existing problems and utilizing predefined rules and templates to create geometric images and problems. However, these approaches often produce data that lacks diversity or is prone to noise. Additionally, the geometric images synthesized by existing methods tend to exhibit limited variation and deviate significantly from authentic geometric diagrams. To overcome these limitations, we propose GeoFM, a novel method for synthesizing geometric data. GeoFM uses formal languages to explore combinations of conditions within metric space, generating high-fidelity geometric problems that differ from the originals while ensuring correctness through a symbolic engine. Experimental results show that our synthetic data significantly outperforms existing methods. The model trained with our data surpass the proprietary GPT-4o model by 18.7% on geometry problem-solving tasks in MathVista and by 16.5% on GeoQA. Additionally, it exceeds the performance of a leading open-source model by 5.7% on MathVista and by 2.7% on GeoQA.