# GeoSketch: A Neural-Symbolic Approach to Geometric Multimodal Reasoning with Auxiliary Line Construction and Affine Transformation

arXiv: https://arxiv.org/abs/2509.22460 · HTML: https://mlsanity.com/paper/2509.22460/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Spatial and embodied reasoning, one of 17 directions in LLM reasoning.
- Builds on 21 of its 38 references that are in the map.
- First posted on arXiv 26 Sep 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/)

## Builds on (most cited first, 21 of 21)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [GPT-4o System Card](https://mlsanity.com/paper/2410.21276/) (2024)
- [Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities](https://mlsanity.com/paper/2507.06261/) (2025)
- [YOLOv11: An Overview of the Key Architectural Enhancements](https://mlsanity.com/paper/2410.17725/) (2024)
- [We-Math: Does Your Large Multimodal Model Achieve Human-like Mathematical Reasoning?](https://mlsanity.com/paper/2407.01284/) (2024)
- [GeoQA: A Geometric Question Answering Benchmark Towards Multimodal Numerical Reasoning](https://mlsanity.com/paper/2105.14517/) (2021)
- [Thinking with Images for Multimodal Reasoning: Foundations, Methods, and Future Frontiers](https://mlsanity.com/paper/2506.23918/) (2025)
- [Multimodal Chain-of-Thought Reasoning: A Comprehensive Survey](https://mlsanity.com/paper/2503.12605/) (2025)
- [Seed1.5-Thinking: Advancing Superb Reasoning Models with Reinforcement Learning](https://mlsanity.com/paper/2504.13914/) (2025)
- [VisuLogic: A Benchmark for Evaluating Visual Reasoning in Multi-modal Large Language Models](https://mlsanity.com/paper/2504.15279/) (2025)
- [GoT: Unleashing Reasoning Capability of Multimodal Large Language Model for Visual Generation and Editing](https://mlsanity.com/paper/2503.10639/) (2025)
- [A Survey of Mathematical Reasoning in the Era of Multimodal Large Language Model: Benchmark, Method & Challenges](https://mlsanity.com/paper/2412.11936/) (2024)
- [SketchAgent: Language-Driven Sequential Sketch Generation](https://mlsanity.com/paper/2411.17673/) (2024)
- [RBench-V: A Primary Assessment for Visual Reasoning Models with Multi-modal Outputs](https://mlsanity.com/paper/2505.16770/) (2025)
- [Draw with Thought: Unleashing Multimodal Reasoning for Scientific Diagram Generation](https://mlsanity.com/paper/2504.09479/) (2025)
- [MATP-BENCH: Can MLLM Be a Good Automated Theorem Prover for Multimodal Problems?](https://mlsanity.com/paper/2506.06034/) (2025)
- [Interactive Sketchpad: A Multimodal Tutoring System for Collaborative, Visual Problem-Solving](https://mlsanity.com/paper/2503.16434/) (2025)
- [VisAidMath: Benchmarking Visual-Aided Mathematical Reasoning](https://mlsanity.com/paper/2410.22995/) (2024)
- [Plane Geometry Problem Solving with Multi-modal Reasoning: A Survey](https://mlsanity.com/paper/2505.14340/) (2025)

## Built on by (most cited first, 6 of 6)

- [V-Thinker: Interactive Thinking with Images](https://mlsanity.com/paper/2511.04460/) (2025)
- [Reasoning or Pattern Matching? Probing Large Vision-Language Models with Visual Puzzles](https://mlsanity.com/paper/2601.13705/) (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)
- [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)

Geometric Problem Solving (GPS) poses a unique challenge for Multimodal Large Language Models (MLLMs), requiring not only the joint interpretation of text and diagrams but also iterative visuospatial reasoning. While existing approaches process diagrams as static images, they lack the capacity for dynamic manipulation - a core aspect of human geometric reasoning involving auxiliary line construction and affine transformations. We present GeoSketch, a neural-symbolic framework that recasts geometric reasoning as an interactive perception-reasoning-action loop. GeoSketch integrates: (1) a Perception module that abstracts diagrams into structured logic forms, (2) a Symbolic Reasoning module that applies geometric theorems to decide the next deductive step, and (3) a Sketch Action module that executes operations such as drawing auxiliary lines or applying transformations, thereby updating the diagram in a closed loop. To train this agent, we develop a two-stage pipeline: supervised fine-tuning on 2,000 symbolic-curated trajectories followed by reinforcement learning with dense, symbolic rewards to enhance robustness and strategic exploration. To evaluate this paradigm, we introduce the GeoSketch Benchmark, a high-quality set of 390 geometry problems requiring auxiliary construction or affine transformations. Experiments on strong MLLM baselines demonstrate that GeoSketch significantly improves stepwise reasoning accuracy and problem-solving success over static perception methods. By unifying hierarchical decision-making, executable visual actions, and symbolic verification, GeoSketch advances multimodal reasoning from static interpretation to dynamic, verifiable interaction, establishing a new foundation for solving complex visuospatial problems.