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MetaReason: Precise Interleaved Multimodal Reasoning via Editing Meta Information for Solving Geometry Problems

Penghao Yin, Haomin Wang, Qihong Tang, Xiaoye Qu, Hongjie Zhang, Xiao-Ping Zhang

arXiv:2608.15006v1cs.CVcs.AIcs.MM

TL;DR

Planar geometry remains difficult for vision-language models because text-only reasoning struggles to represent intermediate visual structures. MetaReason edits structured geometric meta-information through lightweight tools to construct auxiliary lines, achieving state-of-the-art benchmark performance, including 56.1% accuracy on ExamGeo.

  • Problem

    Text-only reasoning makes it difficult for vision-language models to represent intermediate visual structures needed for complex planar geometry.

  • Method

    MetaReason parses geometry into structured meta-information and uses constrained tool calls, supervised fine-tuning, and reinforcement learning to construct auxiliary lines and reason multimodally.

  • Results

    56.1% accuracy on ExamGeo establishes state-of-the-art performance, while MetaReason also outperforms all open-source models on GeoLaux-mini and GeoQA.

  • Takeaways & Limitations

    The results support explicit think-with-image reasoning through precise meta-information manipulation for highly complex, multi-step geometric tasks.

  • Takeaways & Limitations

    Existing visual geometry methods can require coding or image-generation reliability and may lack the precision needed for accurate geometric constructions.

Abstract

from arXiv · show

Although visual reasoning is crucial for solving complex geometry tasks, existing vision-language models rely heavily on text-only reasoning. Some recent methods introduce intermediate visual states to facilitate reasoning, but they are often hindered by inaccurate geometric representations and low rendering fidelity, ultimately leading to unreliable outputs. To address these limitations, we propose MetaReason, a framework for multimodal reasoning in plane geometry that leverages structured meta-information to enable accurate auxiliary-line construction. The framework first parses geometric images into meta-information, performs controllable edits with predefined tools to synthesize high-fidelity visual states, and then conducts reasoning based on these augmented views. To support this framework, we construct TutorGeo, a comprehensive dataset containing 17k image-to-meta conversion samples, 60k text-only reasoning traces, and 60k interleaved multimodal reasoning traces. Using this dataset, we combine supervised fine-tuning and reinforcement learning to develop robust multimodal reasoning capabilities. We also introduce ExamGeo, a benchmark derived from real-world examination problems that enables systematic evaluation across varying difficulty levels. Experimental results demonstrate that MetaReason significantly outperforms existing open-source models and achieves competitive performance against proprietary models.

1 Introduction

MetaReason addresses the difficulty of planar geometry reasoning by editing structured meta-information to construct precise auxiliary lines and support interleaved multimodal reasoning. The paper introduces a trained pipeline, TutorGeo dataset, ExamGeo benchmark, and experiments showing stronger performance than existing baselines.

  • Motivation: Planar geometry remains challenging for vision-language models because it requires fine-grained spatial perception and multi-step reasoning, while auxiliary lines expose latent geometric relationships.Human experts simplify geometry problems by constructing auxiliary lines that provide helpful intermediate visual structures.
  • Method: MetaReason represents geometric figures as JSON-formatted meta-information containing points and shapes, then edits this structure to construct auxiliary lines.A predefined program deterministically renders each meta-information instance, and a toolbox constrains actions to valid geometry operations.
  • Method: MetaReason combines MetaConverter, MetaJudge, and MetaReasoner in a unified planar-geometry pipeline, trained with supervised fine-tuning and reinforcement learning.A dedicated geometry reward mechanism guides tool-calling policies through the large auxiliary-line construction search space.
  • Datasets and benchmark: 137k high-quality instances comprise TutorGeo, while ExamGeo contains 1,000 challenging problems with fine-grained difficulty levels defined by human expert success rates.TutorGeo includes multimodal and text-only reasoning trajectories validated by human experts for logical consistency.
  • Results: Extensive experiments show that MetaReason consistently outperforms strong baselines in planar geometry reasoning.The framework is designed to replace costly image or code generation with lightweight tool invocation for precise auxiliary-line construction.

2 Related Works

Prior work established multimodal geometry benchmarks and introduced visual-state reasoning, including auxiliary-line construction. However, existing benchmarks inadequately assess difficult plane-geometry reasoning, while current visual construction methods face reliability, overhead, and fidelity limitations.

  • Multimodal Geometry Benchmarks: Geometry3K and GeoQA established foundational multimodal geometry reasoning settings, followed by broader benchmarks such as MMMU, MathVista, Math-Vision, and MathVerse.
  • Multimodal Geometry Benchmarks: Existing benchmarks inadequately evaluate plane geometry because general multimodal math datasets contain few such problems, while geometry-specific datasets rarely require auxiliary-line construction.Geometry3K and GeoQA mainly contain relatively straightforward problems.
  • Intermediate Visual Reasoning: Recent systems create intermediate visual states through cropping, bounding boxes, zooming, adaptive focusing, active grounding, and point-based grounding.Examples include OpenAI o3, Chain-of-Focus, DeepEyes, and Point-RFT.
  • Auxiliary-Line Construction: Auxiliary-line methods either render images with external code or generate them through unified models, but coding approaches incur execution and context overhead, while image generators lack precision.Code-based methods include Visual Sketchpad and CodePlot-CoT; unified-model approaches include Zebra-CoT and MathCanvas.

3 MetaReason

MetaReason uses a three-model, interleaved multimodal pipeline that extracts and verifies structured geometric meta-information before iterative tool-based visual sketching. It trains MetaConverter and MetaReasoner with supervised fine-tuning, then further optimizes MetaReasoner with reinforcement learning for multi-step planning and tool use.

  • Pipeline: MetaReason comprises MetaConverter, MetaJudge, and MetaReasoner and processes geometry problems through meta-information extraction, verification, and iterative visual sketching.Image-free inputs bypass extraction and use pure-text reasoning.
  • Pipeline: MetaJudge re-renders extracted meta-information and accepts it only when the reconstruction matches the diagram or obeys the problem’s geometric constraints.Failed verification triggers pure-text chain-of-thought reasoning to prevent error propagation.
  • Representation and tools: The diagram representation uses point coordinates and Segment, Line, and Circle primitives, while sketching tools focus on drawing segments, finding intersections, and drawing perpendicular lines.Tool calls update the JSON representation and invoke the rendering engine to produce new visual sketches.
  • Training: Training uses supervised fine-tuning for structured abstraction, tool use, and reasoning, followed by GRPO reinforcement learning on interleaved multimodal trajectories.MetaConverter learns from image-to-meta pairs; MetaReasoner learns from multimodal and pure-text traces before reinforcement learning.
  • Training: The reward function encourages accuracy and proactive tool use, ranges from 0 to 1.2, and includes α = 0.15 for perfect completion and β = 0.05 for successful sketching-tool use with a fully correct answer.Qwen3-VL-8B-Instruct serves as the reward model.

4 Dataset Construction and Benchmark

This section introduces TutorGeo, a multimodal geometry-reasoning dataset, and ExamGeo, a held-out benchmark for evaluating planar geometry reasoning across difficulty levels. It also describes the automated pipeline for producing structured meta-information, rewritten reasoning, tool calls, and intermediate visual states.

  • Dataset construction: Existing planar-geometry datasets lack programmatic diagram-construction steps and sufficiently detailed reasoning trajectories for training multimodal agents.Their solutions are primarily human-oriented and often omit analysis, logical derivation, and reflective reasoning.
  • Dataset construction: TutorGeo contains 17k image-to-meta conversion data, 60k text-only reasoning data, and 60k interleaved multimodal reasoning data.ExamGeo is a high-quality benchmark derived from real-world examination problems.
  • Dataset construction: TutorGeo transforms raw geometry samples through meta-information extraction, trajectory rewriting, tool-call injection, and visual-state generation.The pipeline uses Gemini-3-Flash for structured image parsing and Qwen3-VL-8B-Instruct for rewriting and identifying auxiliary-line insertion points.
  • Dataset construction: TutorGeo retains text-only reasoning samples, trains image-to-meta parsing from Gemini-3-Flash traces, and uses 6K interleaved samples for reinforcement learning.Text-only samples bypass visual generation, while reinforcement learning targets multimodal reasoning and auxiliary-line planning.
  • Benchmark: ExamGeo contains 1,000 held-out questions grouped into L1–L4 by human accuracy rates of 0.8, 0.6, 0.4, and 0.2, respectively.Lower human accuracy indicates greater difficulty, and GPT-5.2 evaluates final answers because rule-based parsing is inadequate.

5 Experiments

MetaReason-RL achieves strong performance across ExamGeo, GeoLaux-mini, and GeoQA, surpassing open-source baselines and competing with proprietary models. Ablations show that visual reasoning and TutorGeo training are both important, especially for difficult geometry problems.

  • Performance on ExamGeo: 56.1% average accuracy on ExamGeo makes MetaReason-RL state of the art and 7.4 points higher than GPT-5.2’s 48.7%.It reaches 51.6% on L3 and 45.5% on L4, while Gemini-3.1-Pro scores 28.0% and 16.5% on those levels.
  • Performance on GeoLaux-mini and GeoQA: 80.1% accuracy on GeoLaux-mini surpasses GPT-5.2 at 76.9% and Bagel at 44.8%, validating performance on auxiliary-line construction.The passage attributes this result to predefined sketching tools and deterministic rendering.
  • Performance on GeoLaux-mini and GeoQA: 87.5% accuracy on GeoQA improves over the base model’s 62.5% and BAGEL-Canvas’s 82.4%, although Gemini-3.1-Pro reaches 97.1%.MetaReason-RL improves Angle, Area, and Length categories, which together account for 98.6% of GeoQA.
  • Reliability of MetaConverter and MetaJudge: MetaConverter conversion accuracy ranges from 52.5% to 82.0%, while MetaJudge’s acceptance precision and inconsistent rejection both exceed 90%.Both modules are evaluated through human annotation across the three benchmarks.
  • Effect of the MetaReason Framework: Removing visual reasoning reduces GeoQA accuracy by 10.4 points and GeoLaux-mini accuracy by 4.5 points, while ExamGeo gains reach 7.0 and 12.5 points on L3 and L4.The results indicate that intermediate visual construction becomes more important as problem difficulty increases.
  • Effect of TutorGeo Training: Removing TutorGeo training lowers accuracy from 56.1% to 24.1% on ExamGeo and from 80.1% to 46.6% on GeoLaux-mini.The gaps are attributed to the importance of reliable meta-information generation, multi-step planning, and tool invocation.

6 Conclusion · A Definition of Meta-Information

MetaReason combines structured meta-information editing with interleaved multimodal reasoning for geometry, supported by TutorGeo and ExamGeo and trained through SFT and RL. Its meta-information represents geometric images through named points and point-based shapes, enabling precise correspondence between symbolic structures and rendered layouts.

  • 6 Conclusion: MetaReason constructs auxiliary lines by editing structured meta-information with lightweight tool calls.The framework is trained with supervised fine-tuning and reinforcement learning to learn multi-step planning and tool use.
  • 6 Conclusion: 137k samples comprise TutorGeo, while ExamGeo provides difficulty levels based on human accuracy.These resources support training and systematic evaluation of the proposed framework.
  • 6 Conclusion: MetaReason achieves state-of-the-art performance on ExamGeo and GeoLaux-mini.The passage reports this result without specifying the corresponding metric values.
  • A Definition of Meta-Information: Each image’s meta-information has two primary components: points and shapes.Points use unique names and spatial coordinates, while shapes include segments, lines, and circles constructed from points.
  • A Definition of Meta-Information: Rendering points and shapes into a geometric figure demonstrates the mapping from symbolic representations to visual layouts.The visualization illustrates how meta-information captures the underlying structural logic of geometric images.
  • A Definition of Meta-Information: Predefined fields specify geometric elements including points, segments, lines, and circles.The specifications include fields such as coordinates, points, and radius.

B Tool Selection

MetaReason retains three frequently used geometric operations to balance construction coverage with training stability. Its tool set is extensible through new tool definitions, meta-information schemas, and renderer interfaces.

  • Tool selection: MetaReason retains draw_segment, draw_intersection_point, and draw_perpendicular_to_linear as the three most frequently used operations.The broader tool space was reduced because larger action spaces increase prompt length and tool-selection ambiguity during training.
  • Tool selection: The reduced tool set trades construction coverage for training stability rather than imposing an inherent framework limitation.The selection was based on tool-usage distributions in training trajectories.
  • Extensibility: Additional constructions can be supported by adding tool definitions and updating the meta-information schema and renderer interfaces.MetaReason’s central contribution is the combination of structured geometric states, tool-based editing, and deterministic rendering, not a fixed tool set.

C Accuracy of MetaConverter

MetaConverter is evaluated through a human consistency study that tests whether re-rendered diagrams preserve essential geometric information. Lower consistency on ExamGeo reflects its greater problem and diagram complexity, so MetaReason filters inconsistent reconstructions with MetaJudge before visual reasoning.

  • Human consistency study: Three PhD-level annotators independently assess whether re-rendered diagrams preserve geometric structure, point-line relations, and key annotations.The study samples 20% of examples from ExamGeo, GeoLaux-mini, and GeoQA and reports strict diagram-level consistency.
  • Human consistency study: The evaluation reports strict diagram-level consistency for MetaConverter across ExamGeo, GeoLaux-mini, and GeoQA.These results are presented in Table 4.
  • Robustness mitigation: ExamGeo has lower consistency because its problems and geometric diagrams are more challenging and complex.MetaReason uses MetaJudge to filter inconsistent reconstructions before tool-augmented visual reasoning, reducing the impact of inaccurate meta-information.

D Accuracy of MetaJudge

MetaJudge’s reliability is assessed through manual inspection by three PhD-level annotators using acceptance precision and inconsistent rejection rate. High acceptance precision limits inconsistent reconstructions entering reasoning, while remaining errors are usually minor or irrelevant to key geometric relations.

  • Manual evaluation: Three PhD-level annotators manually inspect MetaJudge outputs to evaluate reconstruction reliability.The evaluation uses two complementary metrics.
  • Evaluation metrics: Acceptance precision measures the proportion of accepted samples that remain visually consistent with the original image.It reflects the reliability of samples retained for subsequent reasoning.
  • Evaluation metrics: Inconsistent rejection rate measures how often visually inconsistent reconstructions are correctly rejected.It directly evaluates MetaJudge’s ability to filter inconsistent reconstructions.
  • Error analysis: High acceptance precision keeps few inconsistent reconstructions from entering tool-augmented reasoning, and mistaken acceptances usually involve minor or solution-irrelevant errors.Observed errors include point or line displacements, redundant objects, and inaccuracies unrelated to the solution; these may preserve key geometric relations.

E GeoQA Category Distribution

The GeoQA test set comprises 5,010 examples distributed mainly across Angle and Length categories, while Other accounts for only 1.4%. Accuracy comparisons on Other should therefore be interpreted cautiously because a few predictions can substantially change results.

  • Category distribution: 5,010 test examples comprise 2,745 Angle, 323 Area, 1,873 Length, and 69 Other cases, with Other representing 1.4% of the set.Because Other contains only 69 examples, its accuracy can change substantially with a few different predictions.

F Case Study · G Codes · G.1 Code for Rendering Meta-Information

The case studies illustrate geometry reasoning on ExamGeo and related benchmarks, while the rendering code converts structured metadata into controlled geometry images. The renderer supports points and basic shapes with bounded, equal-aspect visualization and configurable output behavior.

  • F Case Study: The ExamGeo case study presents an angle-bisector configuration with perpendiculars PM and PN and asks which four geometric conclusions are correct.The setup states that BP and AP are angle bisectors, PM ⊥ BE, and PN ⊥ BF.
  • F Case Study: The case-study solutions disagree: one selects ④, another selects ③, and a third selects ①③ or ①②③.These outputs reflect alternative reasoning traces for the same geometry problem.
  • F Case Study: Constructing PD ⊥ AC establishes PM = PN = PD by the equal-distance property of an incenter.The solution identifies P as the incenter of triangle BEC and uses equal perpendicular distances to its sides.
  • F Case Study: Additional case studies cover a folding-stool length problem with ground truth C. 41.6cm and a square-triangle area-ratio problem.The stool problem specifies a 32 cm height and ∠DOB = 100°.
  • G.1 Code for Rendering Meta-Information: The rendering implementation initializes geometry from metadata containing points and shapes, with configurable minimum and maximum image sizes and optional point drawing.The public draw_geometry function returns a rendered image or None on failure, depending on the error-handling option.
  • G.1 Code for Rendering Meta-Information: The renderer computes bounds, selects a preferred Times-like font when available, and creates equal-aspect Matplotlib figures with margins around the geometry.Preferred fonts include Times New Roman, Nimbus Roman, STIX, and P052.

G.2 Implementation Detail of Toolbox · H Prompts

The toolbox represents geometry through named points and typed shapes, with operations for intersections, segment membership, connections, and perpendicular constructions. The prompts separately specify structured figure extraction, reconstruction judging, multimodal sketchpad reasoning, and text-only solution formatting.

  • G.2 Implementation Detail of Toolbox: The toolbox retrieves point coordinates from meta_data and registers newly named points in its points dictionary.These operations are implemented by _get_coord and _add_point.
  • G.2 Implementation Detail of Toolbox: The toolbox connects existing points as lines or segments and appends the resulting shape to meta_data.It validates the linear type and requires both referenced points to exist.
  • G.2 Implementation Detail of Toolbox: The intersection operation computes a point from two linear elements, checks segment membership when required, and rejects missing or non-unique intersections.Parallel or coincident lines trigger an error when the denominator magnitude is below 1e-9.
  • Prompt for MetaConverter: The MetaConverter prompt instructs the model to extract points, coordinates, and geometric relationships from the problem text and image without solving the problem.It prefers integers or simple fractions during reasoning and converts fractions to floating-point coordinates with up to three decimal places.
  • Prompt for MetaConverter: MetaConverter outputs valid JSON containing a points dictionary and typed geometric shapes whose point references, numeric fields, and styles follow explicit schema constraints.Supported objects include segments, infinite lines, and circles; referenced points must be defined and shapes must reflect the original image.
  • Prompt for MetaJudge: The MetaJudge compares reconstructed and question images topologically, then verifies explicit textual constraints when structural differences exist.It labels rational reconstructions as \boxed{False} and irrational ones as \boxed{True}.
  • Prompt for MetaReasoner to Implement Multimodal Reasoning: The multimodal reasoner is instructed to construct auxiliary geometry only when necessary, continue reasoning from the updated figure, and format tool calls and conclusions in designated tags.Its sketchpad supports drawing segments, intersections, and perpendiculars while preventing newly constructed points from overwriting existing ones.
  • Prompt for MetaReasoner to Implement Text-only Reasoning: The text-only reasoner is prompted to perform rigorous geometric analysis, cautiously construct auxiliary lines conceptually, and place the final answer or proof conclusion inside boxed answer tags.The prompt emphasizes detailed solution steps and accurate conclusions based on the image and problem description.
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