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When Does Auxiliary Modality Matter in Solving Geometric Problems? A Comprehensive Study of Textual, Formal, and Visual Modalities

Paper · 2026 · Hyuk Namgoong, Jeesu Jung, Yerim Han et al.

Summary

Geometry problems require a model to connect what the text says with what the diagram shows, and multimodal LLMs often misread diagrams. Earlier work added diagram parsers, but no study had compared different kinds of extra input, alone and combined. This short paper fills that gap. It tests four auxiliary modalities: formal diagram facts in CDL, a natural-language version of them (T_CDL), exhaustive diagram descriptions (DES), and synthesized variant diagrams (IMG). These are added to the problem statement and diagram and evaluated on the FormalGeo7K and Geometry3K test sets with open-source and closed multimodal LLMs. Formal inputs give small gains. IMG has almost no effect. DES splits the models: it substantially helps weaker open-source models that struggle to parse diagrams, but it tends to hurt stronger closed models, probably because generated descriptions contain factual mistakes. The study's main lesson is that extra input can be useful information or harmful noise depending on the model's own visual competence. The authors therefore suggest selective, gated use of auxiliary inputs.

Key points

  • Modalities and how they were built:
    • CDL comes from the Diagram Formalizer (Zhang et al., 2025), restricted to collinearity and cocircularity, with a calibrated variant used for evaluation.
    • T_CDL is a rule-based rendering of CDL into plain text.
    • DES and IMG were produced by GPT-o4-mini. For IMG, the model generated Python drawing code with rotations, flips and affine transforms, retrying when rendering failed.
  • Models tested:
    • Closed: GPT-o4-mini and Gemini 2.5-flash, with and without "think".
    • Open-source: Qwen-2.5-VL 72B-Instruct, Llama-4-Scout 17B-16E-Instruct and Gemma-3 27b-it.
  • Single-modality effects:
    • T_CDL improved accuracy by at most +3.6% on FormalGeo7K and +2.1% on Geometry3K.
    • IMG changed results only marginally, by about +0.2% on average.
  • Effect of DES:
    • Open-source models gained up to +15.0%.
    • Closed models lost up to −6.8% on FormalGeo7K, a dataset whose problem text already carries most of the needed information.
  • Combinations and limitations:
    • For closed models, adding CDL and T_CDL alongside DES narrowed one drop on FormalGeo7K from −4.2% to −1.9%.
    • The study covers only planar geometry and measures only final-answer accuracy.
    • The generated DES and IMG were not checked by humans.

Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.

 Institutions

🎓Chungnam National UniversityDaejeon · South Korea5 works · 13 citations

 Fields, topics and concepts

📚 Shelves 7The field of science this work belongs to (OpenAlex fields).Computer Science · 16,667 worksArtificial Intelligence · 16,194 worksArts and Humanities · 16,039 worksLanguage and Linguistics · 15,891 worksComputer Vision and Pattern Recognition · 2,869 worksMathematics · 177 worksApplied Mathematics · 46 works

Cite

Cite as Hyuk Namgoong, Jeesu Jung, Yerim Han and Sangkeun Jung. 2026. When Does Auxiliary Modality Matter in Solving Geometric Problems? A Comprehensive Study of Textual, Formal, and Visual Modalities. Proceedings of the 19th Conference of the European Chapter of the Association for Computational Linguistics (Volume 2: Short Papers).

The paper itself, not this page. BibTeX from the library's record of the work. DOI 10.18653/v1/2026.eacl-short.4

@inproceedings{namgoong2026when,
  title = {{When Does Auxiliary Modality Matter in Solving Geometric Problems? A Comprehensive Study of Textual, Formal, and Visual Modalities}},
  author = {Hyuk Namgoong, Jeesu Jung, Yerim Han and Sangkeun Jung},
  booktitle = {Proceedings of the 19th Conference of the European Chapter of the Association for Computational Linguistics (Volume 2: Short Papers)},
  year = {2026},
  doi = {10.18653/v1/2026.eacl-short.4},
  url = {https://doi.org/10.18653/v1/2026.eacl-short.4}
}

References

4 references, 1 of them with a card of ours. ↗ leads to the work, «find» searches for one we have no id for yet.

  1. Meta AI. 2025. The llama 4 herd: Launching Llama 4 Scout. https://ai.meta.com/blog/ llama-4-multimodal-intelligence/. Blog post. Accessed 18 May 2025. ↗
  2. A Experiments Setting OpenAI. 2025. Gpt-4o mini: Advancing cost-efficient intelligence. https://openai.com/index/ gpt-4o-mini-advancing-cost-efficient-intelligence/. Blog post. Accessed 18 May 2025. Dataset Setting FormalGeo7K is an Olympiad-style geometry dataset comprising geometric diagrams, prob… ↗
  3. Yicheng Pan, Zhenrong Zhang, Pengfei Hu, Jiefeng Ma, Jun Du, Jianshu Zhang, Quan Liu, Jianqing Gao, and Feng Ma. 2025. Enhancing the geometric problem-solving ability of multimodal llms via symbolic-neural integration. Preprint, arXiv:2504.12773. — [arXiv:2504.12773](https://arxiv.org/abs/2504.12773) find
  4. Minjoon Seo, Hannaneh Hajishirzi, Ali Farhadi, Oren Etzioni, and Clint Malcolm. 2015. Solving geometry problems: Combining text and diagram interpretation. In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, pages 1466–1476, Lisbon, Portugal. Association for Co…

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