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Synthesizing Multimodal Geometry Datasets from Scratch and Enabling Visual Alignment via Plotting Code

Haobo Lin, Tianyi Bai, Chen Chen, Jiajun Zhang, Bohan Zeng, Wentao Zhang, Binhang Yuan

arXiv:2602.18745v1cs.CVcs.AI

TL;DR

Multimodal geometry models still struggle to recover geometric structure from complex diagrams, despite the importance of that structure for reasoning. The paper builds GeoCode through verified synthesis and trains models to predict plotting code as an explicit alignment target. GeoCode is more structurally complex and difficult than existing benchmarks, and models trained on it improve consistently across public geometry benchmarks, including out-of-distribution settings.

  • Problem

    Multimodal models struggle to recover geometric structure and maintain intermediate-step consistency on complex constructions, while existing datasets and answer- or text-level supervision leave structure recovery weakly constrained.

    Source: §1 · pp. 1–2

  • Method

    The paper synthesizes GeoCode from symbolic seeds through grounded instantiation and verification, and uses plotting-code prediction to supervise geometric structure recovery from diagrams.

    Source: §1 · pp. 2–3

  • Results

    GeoCode has higher structural complexity and reasoning difficulty than existing benchmarks, and models trained on it improve consistently across multiple public geometry benchmarks, including out-of-distribution evaluation.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Plotting code provides an explicit alignment target that makes geometric perception a supervised component rather than an implicit representation.

    Source: §1 · p. 3

    Instantiation relies heavily on existing LLM reasoning, discards approximately 50% of samples, and requires extra computation to validate results.

    Source: I Limitations · p. 33

Abstract

from arXiv · show

Multimodal geometry reasoning requires models to jointly understand visual diagrams and perform structured symbolic inference, yet current vision--language models struggle with complex geometric constructions due to limited training data and weak visual--symbolic alignment. We propose a pipeline for synthesizing complex multimodal geometry problems from scratch and construct a dataset named \textbf{GeoCode}, which decouples problem generation into symbolic seed construction, grounded instantiation with verification, and code-based diagram rendering, ensuring consistency across structure, text, reasoning, and images. Leveraging the plotting code provided in GeoCode, we further introduce code prediction as an explicit alignment objective, transforming visual understanding into a supervised structured prediction task. GeoCode exhibits substantially higher structural complexity and reasoning difficulty than existing benchmarks, while maintaining mathematical correctness through multi-stage validation. Extensive experiments show that models trained on GeoCode achieve consistent improvements on multiple geometry benchmarks, demonstrating both the effectiveness of the dataset and the proposed alignment strategy. The code will be available at https://github.com/would1920/GeoCode.

1 Introduction

The paper identifies visual-to-symbolic structure recovery as a major bottleneck in multimodal geometry reasoning and proposes GeoCode, a verified synthesis pipeline with plotting-code supervision. It reports greater structural complexity and reasoning difficulty than existing benchmarks and consistent gains on multiple public geometry benchmarks, including out-of-distribution settings.

1 Introduction

  • The paper identifies faithful recovery of geometric structure from diagrams, rather than downstream deduction alone, as the dominant multimodal bottleneck.Formal solvers can approach IMO-level problems when given accurate symbolic representations, while multimodal models struggle with structure recovery and intermediate-step consistency.Source: §1 · pp. 1–2
  • Existing datasets offer limited structural diversity and reasoning depth, while answer- or text-level supervision leaves diagram-based relation recovery weakly constrained.These gaps may leave models relying on linguistic correlations or dataset biases rather than explicitly reconstructing geometric relations.Source: §1 · p. 2
  • Geometry's symbolic solvability and coordinate-based verifiability make it possible to synthesize problems with correctness checks and cross-modal consistency.This opportunity remains underexploited in existing multimodal geometry datasets.Source: §1 · p. 2
  • GeoCode's pipeline constructs symbolic seeds, instantiates them with numerical parameters and language, then generates coordinate-level constructions and validates retained samples across modalities.Validation includes semantic and geometric checks plus visual-quality filtering.Source: §1 · pp. 2–3
  • Plotting-code prediction makes geometric perception an explicit supervised target by requiring reconstruction of constructions and spatial relations from diagrams.The paper presents GeoCode as an 18K dataset with diagrams, solutions, and plotting code.Source: §1 · p. 3
  • GeoCode yields higher structural complexity and reasoning difficulty than existing benchmarks, while trained models improve consistently across multiple public geometry benchmarks, including out-of-distribution settings.The paper also reports that ablations show plotting-code supervision has clear advantages over text-only alignment.Source: §1 · p. 3

2 Related Work

Related work covers multimodal reasoning methods and geometry-specific approaches, while identifying dataset quality and explicit visual–symbolic alignment as unresolved challenges. The paper's synthesis pipeline is organized into three verifiable stages.

2 Related Work

  • Multimodal reasoning work uses intermediate reasoning traces and reinforcement-learning alignment methods to improve reasoning behavior on complex benchmarks.The passage names CoT, PPO, and GRPO as examples.Source: §2 · p. 3
  • Geometry-solving approaches include alignment-oriented, synthetic-data, and solver-driven methods, but large-scale complex datasets with strict cross-modal consistency and explicit structure reconstruction remain limited.The paper identifies answer- or text-only supervision as a key gap in existing alignment mechanisms.Source: §2 · p. 3
  • The proposed synthesis framework separates symbolic seed generation, grounded instantiation, and visualization with textual debiasing.The stages respectively handle relational structure, numerical grounding and meta-code, and diagram rendering.Source: Figure 2 · p. 4

3 Method

The method separates abstract geometric structure from numerical grounding and visual presentation, then uses plotting code to supervise recovery of diagrams' underlying structure. Semantic, geometric, and visual checks support the retained instances.

3.1 Generation Pipeline

  • The generation pipeline factorizes synthesis into symbolic seed generation, grounded instantiation with verification, and visualization with textual debiasing.This modular decomposition separates logical structure, metric realization, and perceptual presentation.Source: §3.1 · p. 4, §3 · p. 4

3.1.1 Seed Generation

  • Seed generation samples relational predicates, uses AlphaGeometry dependency graphs to find non-trivial proof targets, and selects seeds with both structural breadth and deductive depth.Selected seeds contain premises, ordered proof steps, and target conclusions, but no numerical or coordinate information.Source: §3.1.1 · pp. 4–5

3.1.2 Instantiation

  • Instantiation turns symbolic seeds into numerical problems, language statements, reasoning traces, answers, and geometric construction programs.An Instructor generates the grounded problem and reasoning; a separate Coder derives constructions from the problem statement without access to the reasoning trace or final answer.Source: §3.1.2 · p. 5
  • Semantic checks assess problem coherence, reasoning, and answers, while coordinate-based geometric checks test declared constraints and metric answers; invalid samples are discarded.This two-stage verification controls consistency across modalities.Source: §3.1.2 · pp. 5–6

3.1.3 Visualization

  • Visualization renders parsed construction code into diagrams and filters overlaps or ambiguous layouts; textual debiasing removes diagram-inferable cues from questions.Reasoning traces are rewritten to reflect interactions between visual observations and textual premises, requiring extraction of relations from the image.Source: §3.1.3 · pp. 6–7

3.2 Plotting Code as Explicit Alignment

  • Plotting code explicitly represents geometric entities, constructions, and visual relations, so models predict the scene structure before reasoning and answering.The paper reports that code-based alignment couples visual perception and symbolic reasoning more strongly than answer-only or text-only supervision.Source: Figure 3 · p. 6, §3.2 · p. 7

4 Experiments

Experiments evaluate GeoCode’s quality, its transfer to public geometry benchmarks, and plotting-code supervision as an alignment strategy. Models trained on GeoCode outperform baselines on most public benchmarks, while code supervision produces stronger alignment gains than caption supervision.

4.1 Implementation Details

  • GenesisGeo generates and selects symbolic seeds; GPT-OSS-120B handles instantiation through plotting-code generation, followed by OpenCV rendering and visual filtering.Experiments use Qwen3-VL-7B-Instruct and Qwen2.5-VL-7B-Instruct backbones, with LoRA SFT and standard GRPO for reinforcement learning.Source: §4.1 · p. 8

4.2 Quality and Difficulty of GeoCode

  • GeoCode combines diverse geometric relations, passes semantic and coordinate checks, and shows greater structural richness and difficulty than commonly used data.The dataset contains 18k problems: 13k numerical and 5k proof-based; random manual inspection of 1% found no incorrect or inconsistent cases.Source: §4.2 · p. 8, Table 1 · p. 8, Table 2 · p. 8

4.3 Effectiveness on Test-mini and Public Benchmarks

  • Models trained on GeoCode outperform baselines on most public geometry benchmarks, including Qwen3-VL-Instruct’s +11.5 on OlympiadBench.For general vision–math benchmarks, evaluation uses plane-geometry subsets.Source: Table 3 · p. 9, §4.3 · p. 9
  • 17.9% to 37.7% for Qwen3-VL-Instruct and 4.5% to 20.5% for Qwen2.5-VL-Instruct are the Base-to-GRPO Test-mini accuracy changes after SFT and GRPO.GRPO rewards only answer correctness and output format, without intermediate reasoning or plotting-code supervision.Source: Table 4 · p. 9, §4.3 · p. 9

4.4 Ablation Study on Pipeline Components

  • Of 35,297 selected seeds, 18,176 (51.5%) pass all later gates, which reject semantic, geometric, code-execution, and visual-quality failures.Seed selection removes structurally trivial candidates; subsequent checks address distinct failure types.Source: §4.4 · pp. 9–10, Table 5 · p. 10

4.5 Plotting Code Enables Stronger Visual Alignment

  • Plotting-code supervision yields the strongest alignment gains on Test-mini and Geometry3K; captions add only marginal gains over no alignment.Textual debiasing brings additional gains mainly when combined with code-based supervision.Source: §4.5 · p. 10, Table 6 · p. 10
  • Solving accuracy rises with segment-level F1 for predicted versus ground-truth plotting code, associating structural recovery quality with reasoning performance.Test-mini samples are grouped into four equal-sized F1 bins.Source: §4.5 · pp. 10–11, Figure 5 · p. 11

5 Conclusion

The paper presents GeoCode as a dataset built through a validated synthesis pipeline and uses its plotting code as an alignment target. The authors report stronger visual grounding than text-based supervision and improved performance across geometry benchmarks.

5 Conclusion

  • The pipeline synthesizes multimodal geometry problems from scratch and validates consistency among symbolic structures, text, reasoning traces, and diagrams.Generation is decoupled into symbolic seed construction, grounded instantiation, and code-based visualization.Source: §5 · p. 11
  • GeoCode pairs geometry problems with diagrams, solutions, and plotting code, providing supervision for both perception and reasoning.Source: §5 · p. 11
  • Plotting-code prediction encourages recovery of geometric constructions and yields more robust visual grounding than text-based supervision, with improved performance across geometry benchmarks.Source: §5 · p. 11

Impact Statement

The paper frames GeoCode as a geometry-synthesis and visual-alignment contribution with potential research and education uses. It contrasts its staged generation, code-based alignment, and multi-level validation with prior approaches, then points to broader applications.

Impact Statement

  • The authors identify support for multimodal-reasoning research, geometry-learning tools, and interpretable visual-symbolic models as potential positive impacts.They say the data are synthetically generated and contain no personal data, sensitive content, or real-world imagery.Source: Impact Statement · p. 12
  • The authors note a general risk of academic dishonesty or automated answer generation, while saying it is not specific to this method and that they foresee no significant contribution-specific negative societal consequences.They describe research and education as the work’s primary intended uses.Source: Impact Statement · p. 12

A Discussion

  • Prior synthesis ranges from dataset augmentation and sampled configurations to symbolic search and code-generated figures validated by A/B testing [23] [25] [26] [27].The paper distinguishes GeoCode through its three-stage factorization and code-based alignment [23] [21] [22].Source: A Discussion · p. 16
  • GeoCode decomposes synthesis into basic structure, instance information, and visualization code, and uses plotting code for alignment with multi-level semantic, numerical, and visual checks.The authors say this enables more diverse structures and leverages LLM priors to generate interesting problems.Source: A Discussion · p. 16
  • The authors see future directions in applying synthesized data to Thinking With Images and in extending code-based alignment to bridge visual perception and symbolic reasoning.They also mention potential supervision and reinforcement-learning applications.Source: A Discussion · p. 16

B A Closer Look at Plotting Code as Alignment Signal

The diagnostic analyses associate plotting-code supervision’s gains with recovering diagram structure, especially constraints, rather than merely producing parseable code. Segment-level structural recovery and annotation fidelity track solving accuracy, while parsing success alone has little relationship to it.

Why Annotations Matter

  • Structural correctness of predicted annotations is strongly correlated with solving performance, whereas successful parsing shows little correlation.The figure contrasts parsing success with annotation fidelity as diagnostic signals.Source: Figure 6 · p. 17

Segment-Level Structural F1

  • Higher segment-level F1 is consistently associated with higher solving accuracy, and the metric evaluates recovered diagram relations independently of reasoning wording or answer format.Samples were ranked by segment-level F1 and split into four equal-sized groups.Source: B A Closer Look at Plotting Code as… · p. 17

Parsing Is Not the Bottleneck

  • 219 cases were parsable (81.7%); solving accuracy was 37.9% for parsable cases and 36.7% for non-parsable cases.The authors characterize the accuracy difference as negligible and argue against a format-only explanation.Source: B A Closer Look at Plotting Code as… · pp. 17–18

Why Annotations Matter

  • The annotation analysis focuses on 189 Test-mini instances with explicit annotations and assesses whether predictions are fully correct or partially or fully incorrect.The diagnostic asks whether code supervision teaches well-formed output alone or reconstruction of geometric structure.Source: B A Closer Look at Plotting Code as… · pp. 16, 18

Summary

  • Plotting-code supervision is associated with improved structural reconstruction, especially constraint-bearing relations, rather than gains from enforcing a rigid output template.The analyses connect segment recovery and annotation fidelity with solving performance, while finding little explanatory value in parsing success alone.Source: B A Closer Look at Plotting Code as… · p. 18

C Training and Evaluation Details

Training uses supervised fine-tuning on synthetic geometry examples, followed by GRPO with grouped response rewards. The specified objectives supervise plotting code, reasoning traces, and answers while rewarding format and answer correctness.

C Training and Evaluation Details

C.1 Supervised Fine-Tuning

  • SFT uses cross-entropy training on 18k synthetic geometry samples for two epochs, with a learning rate of 1.0 × 10^-5 and effective batch size 32.The model predicts plotting code, reasoning traces, and final answers from the input prompt and image.Source: §C.1 · pp. 18–19

C.2 Reinforcement Learning with GRPO

  • GRPO groups multiple responses for policy updates and uses a reference-policy KL penalty with coefficient β = 0.01.Training runs for one epoch, sampling four responses per prompt for reward estimation.Source: §C.2 · p. 19
  • The composite GRPO reward combines binary format correctness with binary answer correctness, with answer matching allowing numerical tolerance.The required output has plotting-code, reasoning-trace, and final-answer blocks in that order.Source: §C.2 · p. 19

C.3 Evaluation Protocol

Evaluation uses geometry-focused subsets for non-geometry-specific benchmarks and a standardized, deterministic answer-matching procedure. The supplied GRPO curves show different reward-growth patterns for the two backbones.

C.3 Evaluation Protocol

  • For non-geometry-specific datasets, evaluation subsets follow original annotations and include plane-geometry problems; Table 7 summarizes their sizes.The evaluation includes only solution-type problems requiring solution generation.Source: §C.3 · pp. 19–20, Table 7 · p. 20
  • Qwen3-VL’s GRPO reward curve improves more stably and monotonically, while Qwen2.5-VL’s grows more slowly with larger fluctuations.The figure describes the former pattern as indicating stronger geometric reasoning and better alignment with the hard, structurally complex data.Source: Figure 7 · p. 20
  • All models are evaluated with temperature τ = 0, a shared system prompt, identical answer extraction and matching procedures, and a 10^-6 numerical tolerance.Answers are parsed from <answer> or <box> tags and processed with LaTeX parsing and SymPy symbolic evaluation.Source: §C.3 · p. 20

D Effect of Backbone Models

Qwen3-VL shows steadier optimization and stronger geometric reasoning than Qwen2.5-VL, while both benefit from GRPO on Test-mini. The comparison also reports consistent gains from text-seeded augmentation on GeoQA for both backbones.

D Effect of Backbone Models

  • Qwen2.5-VL improves from 13.1% to 20.5% on Test-mini with GRPO, while Qwen3-VL shows steady reward growth and converges to a high, stable plateau.Qwen2.5-VL's reward curve grows more slowly and has greater variance; Qwen3-VL outperforms it on final Test-mini performance.Source: D Effect of Backbone Models · pp. 20–21
  • Qwen3-VL's specialized STEM and geometry pretraining is associated with more stable GRPO optimization, whereas Qwen2.5-VL has slower, higher-variance reward growth.The authors identify backbone selection as important for geometry and describe the generated data as a probe that distinguishes backbone capabilities.Source: D Effect of Backbone Models · pp. 20–21
  • Text-seeded augmentation yields consistent gains for both backbones on GeoQA, compared with pretrained-only and original-synthetic-data training regimes.The table caption defines Baseline as pretrained backbones, OOD as training only on the original synthetic dataset, and Ours as adding text-seeded synthetic data.Source: Table 8 · p. 21

E Extensibility of Our Pipeline

The pipeline can use structured textual geometry problems as seeds, then instantiate and visualize variants. This text-to-structure route is presented as a way to align synthetic data more closely with real benchmark structures while retaining automatic validation.

E Extensibility of Our Pipeline

  • GeoQA's structured textual problems expose entities and constraints suitable as structural backbones, unlike the relatively sparse descriptions in Geometry3K.The authors use GeoQA-style statements as inputs for text-seeded synthesis.Source: E Extensibility of Our Pipeline · p. 21
  • The extended pipeline reuses existing textual problem stems as high-level seeds, then applies instantiation and visualization to generate variants with coordinates and diagrams.The authors say this text-to-structure augmentation bridges part of the domain gap by aligning synthetic combinatorial structure with real benchmarks, while retaining automatic validation and visualization.Source: E Extensibility of Our Pipeline · p. 21
  • On GeoQA, the authors compare pretrained-only, original-synthetic-data, and text-seeded augmentation regimes; the table caption reports consistent gains for both backbones.The caption characterizes the gains as indicating adaptation to existing geometry corpora and a narrower domain gap.Source: E Extensibility of Our Pipeline · p. 21
  • The authors conclude that the solver-driven pipeline can accept real-world textual problems as structural inputs, with instantiation and visualization operating independently of seed generation.The text-seeded examples retain geometric consistency through the same verification mechanisms.Source: E Extensibility of Our Pipeline · p. 21

F Example Data from Our Pipeline

The examples and pipeline description show how GeoCode generates structurally rich diagrams from both symbolic seeds and existing text. The detailed procedure separates seed generation, instantiation, and visualization, then numerically checks constraints, answers, and annotations before retaining samples.

F Example Data from Our Pipeline

  • Generated problems average 10 predicates, with a long-tailed mix of common and less frequent relations; most contain multiple entities and constraints.The authors report greater structural complexity than existing geometry datasets.Source: F Example Data from Our Pipeline · p. 22, Figure 8 · p. 22
  • Fully synthetic examples include dense constructions, overlapping circles, complex intersections, and interwoven constraints that create multi-step reasoning chains.The examples span relations such as parallelism, perpendicularity, cyclic quadrilaterals, similarity, and congruence.Source: F Example Data from Our Pipeline · p. 22, Figure 9 · p. 23
  • Text-seeded examples show that the same instantiation and visualization stages can process existing problem statements while maintaining geometric consistency.The figure presents 20 diagrams generated from text-seeded synthesis.Source: F Example Data from Our Pipeline · p. 22, Figure 10 · p. 24

G Detailed Pipeline and Algorithms

  • The pipeline samples predicates, builds and filters dependency graphs, selects subgoals, obtains problem statements and reasoning, grounds coordinates, and renders diagrams with image-quality filtering.The stages are symbolic seed generation, instantiation, and visualization.Source: G Detailed Pipeline and Algorithms · p. 25

G.1 Coordinate Verification Mechanism

  • For proof problems, conclusions such as equalities are converted into numerical expressions expected to evaluate to zero using generated coordinates.The method also checks declared geometric constraints and compares evaluated numerical answers with the stated answer.Source: §G.1 · p. 26
  • Annotation checks compare marked lengths, right angles, and angle measures with computed coordinates; any sample failing verification is discarded.Retained instances are intended to preserve numerical consistency across symbolic constraints, coordinates, and diagrams.Source: §G.1 · p. 27

G.2 Plotting Code Format Specification

This section specifies GeoCode’s structured plotting-code representation for rendering verified geometry and supervising visual–symbolic alignment, including its encoded geometric elements and target quantities. It also describes seed-generation rules for triangle congruence and similarity, with predicates for correspondences and resulting relations.

G.2 Plotting Code Format Specification

  • The schema encodes labeled 2D point coordinates and straight segments as endpoint-label pairs, requiring distinct point coordinates.Coordinate pairs contain two numeric values, and each segment connects its two specified points.Source: §G.2 · p. 27
  • Circles can be specified by numeric radius, center-to-point radius, diameter, or three points, and each requires a unique ID.Circle IDs are distinct from point labels and are used to reference circles in DSL expressions.Source: §G.2 · p. 28
  • For training, circular entities use a center-radius format and models regress only the integer part of the radius.Although the generation pipeline supports circles defined by diameter or three points, the renderer computes a deterministic coordinate-level radius for plotting.Source: §G.2 · p. 28
  • Annotations encode right angles, stated segment lengths, and stated angle measures, and must reflect relations explicitly given in the problem.Lengths and angle measures may be numeric strings or expressions; angles may also use radian expressions with π.Source: §G.2 · p. 28
  • Quantities are DSL expressions for values to compute or verify; proof expressions use left minus right and should equal zero when the claim is true.Examples include an angle expression for a computation and a difference of segment lengths for a proof.Source: §G.2 · pp. 28–29
  • Identical plotting code produces identical diagrams, making the code an explicit supervision target for visual–symbolic alignment.The format is designed to preserve geometric information while connecting the generated problem statement, underlying structure, and visual representation.Source: §G.2 · pp. 27, 29

G.3 Seed Generation

  • The direct SSS rule derives contri A B C P Q R from three corresponding side congruences, ncoll A B C, and sameclock A B C P Q R.Source: §G.3 · p. 32
  • The reverse SSS rule derives contrir A B C P Q R using the same side congruences and non-collinearity, but sameclock A B C P R Q.Source: §G.3 · p. 32
  • The direct SAS rule lists two side congruences, an included-angle equality, non-collinearity, and an orientation condition for triangle congruence.Source: §G.3 · pp. 32–33
  • The reverse SAS rule is specified with two side congruences, an included-angle equality, non-collinearity, and an orientation condition.Source: §G.3 · p. 33
  • The direct similarity-without-scaling rule derives contri A B C P Q R from two corresponding side ratios, a side congruence, non-collinearity, and matching orientation.Source: §G.3 · p. 33
  • The reverse similarity-without-scaling rule uses the corresponding ratios and side congruence with non-collinearity and reversed orientation to derive contrir A B C P Q R.Source: §G.3 · p. 33

H API Usage

Section 4.3 evaluates generated-problem difficulty using Gemini-2.5-Pro and GPT-5 through OpenRouter, plus locally served Qwen3-VL-32B-Thinking, with consistent prompts across models.

H API Usage

  • Gemini-2.5-Pro and GPT-5 were accessed through the OpenRouter API for difficulty evaluation.Source: H API Usage · p. 33
  • Qwen3-VL-32B-Thinking was served locally for the evaluation.Source: H API Usage · p. 33
  • System and user prompts remained consistent across models to support fair and correct assessment.Source: H API Usage · p. 33

I Limitations

This section describes GeoCode’s limitations and specifies prompt rules for computation, annotation, validation, diagram-quality screening, and image-grounded captions. It notes limitations in instantiation efficiency, evaluation scope, and coverage of geometry and diagram styles.

I Limitations

  • About 50% of samples are discarded during instantiation with GPT-OSS-120B, and LLM hallucinations require costly validation.The authors expect stronger reasoning models to improve efficiency.Source: I Limitations · p. 33
  • Experiments primarily use Qwen2.5VL and Qwen3VL, so transferability to other VLM backbones remains insufficiently validated.The authors anticipate their data being used to train other VLM series.Source: I Limitations · p. 33
  • The dataset covers only planar Euclidean geometry and idealized textbook-style images, not higher-dimensional geometry or handwritten, sketch, and real exam styles.The planar restriction stems from the structure generated during Seed Generation.Source: I Limitations · p. 33

LLM Generator Prompt (Computation)

  • Requested quantities must use allowed DSL expressions rather than numerical answers, usually one expression per problem and with circle functions taking a circle ID as their first argument.The allowed forms cover point and polygon measurements, angles between lines, and circle quantities; simple arithmetic can combine the basic predicates.Source: LLM Generator Prompt (Computation) · pp. 42–43
  • For proof problems, the prompt converts each claimed equality into a left-minus-right DSL expression expected to evaluate to zero, with multiple expressions allowed for multiple equalities.Examples encode equal lengths, angles, ratios, parallelism, perpendicularity, and collinearity as verification expressions.Source: LLM Generator Prompt (Computation) · pp. 44–45
  • The prompt combines coordinate construction, text-only annotations, and DSL quantity generation in one executable Python program, while prohibiting final-answer output and file or network access.Coordinates must satisfy stated geometric relationships; annotations must not add information inferred through reasoning.Source: LLM Generator Prompt (Computation) · p. 46
  • Annotation rules prohibit adding properties inferred from geometric facts and decimal measurements not explicitly given, advising omission rather than incorrect annotation.The prompt gives examples including equal base angles for an isosceles triangle and equal segments for a midpoint.Source: LLM Generator Prompt (Computation) · p. 47
  • Requested quantities must use allowed DSL expressions that represent what to find, not calculated numerical answers.The prompt requires clear, parseable expressions based on specified predicates and a fixed output dictionary structure.Source: LLM Generator Prompt (Computation) · pp. 48–49
  • Mathematical prompts assess problem self-consistency, condition completeness and sufficiency for solvability, and consistency of the cot and answer; plotting-code prompts assess correctness, completeness, consistency, drawability, and reasonableness.An evaluation output format includes a true/false passed field, a brief reason, and a score from 0 to 100.Source: LLM Generator Prompt (Computation) · pp. 50–51
  • Problem rewriting removes visual or topological definitions while retaining geometric constraints and metrics; a second pass deletes conditions already represented in annotations.Equivalent forms such as perpendicularity and a right-angle annotation are treated as duplicates, while descriptions that cannot be fully represented by annotations are retained.Source: LLM Generator Prompt (Computation) · pp. 52, 54
  • The VQA rewriting prompt allows geometric facts defined in plotting code but prohibits treating properties absent from that code as visual facts.It directs the model to rewrite the original reasoning as compact, sequential steps using only code-defined geometry, implicit problem conditions, and general Euclidean knowledge.Source: LLM Generator Prompt (Computation) · p. 56
  • Image suitability is judged from the visual content of the image itself.The prompt asks whether the image is suitable as an illustration for a geometry problem.Source: LLM Generator Prompt (Computation) · p. 57
  • The prompt rejects diagrams when severe clarity, occlusion, readability, or layout problems affect understanding; otherwise, the image passes.It requires a brief explanation and a JSON pass/fail output.Source: LLM Generator Prompt (Computation) · p. 57
  • The caption prompt requests an objective, complete, non-inferential description based on visible image content, using plotting code only as a structural reference.It instructs the generator not to mention code elements that are not visible in the image.Source: LLM Generator Prompt (Computation) · p. 58
  • The caption must not infer unstated geometric relationships, complete missing figure information, use problem context, or introduce hallucinations.Descriptions are restricted to points, lines, circles, and annotations that are actually visible.Source: LLM Generator Prompt (Computation) · p. 58
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