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Open Access

ARTICLE

KG-HoT: Knowledge-Grounded Hybrid Chain-of-Thought for Geometry Problem Solving

Meihuizi Jia1,*, Hongyan Ran1, Shanshan Li2
1 School of Artificial Intelligence and Computer Science (School of Software), Northwest Normal University, Lanzhou, China
2 Beijing Jinghang Research Institute of Computing and Communication, Beijing, China
* Corresponding Author: Meihuizi Jia. Email: email

Computers, Materials & Continua https://doi.org/10.32604/cmc.2026.080333

Received 06 February 2026; Accepted 06 May 2026; Published online 29 May 2026

Abstract

Large language models (LLMs) have demonstrated considerable ability in solving various tasks via Chain-of-Thought (CoT) prompting, which has precipitated extensive research into their application for complex mathematical reasoning problems. However, current research on mathematical reasoning with CoT predominantly focuses on textual mathematical tasks, such as math word problems, while paying limited attention to multimodal geometric scenarios. To bridge this gap, we propose KG-HoT, a model that harnesses the generative and comprehension capabilities of Multimodal large language models (MLLMs) to enhance complex geometric problem-solving in multimodal systems. Our knowledge-grounded approach enables MLLMs to generate hybrid chains-of-thought operating on dual tracks—language-based reasoning and program-based reasoning—which serve as teaching signals for smaller models. Furthermore, we design an instruction tuning framework that trains these dual reasoning tracks collaboratively within a unified architecture, enabling mutual enhancement and efficient knowledge distillation for complex geometric problem solving. Extensive experimental results demonstrate that KG-HoT achieves superior performance compared to existing approaches on multiple geometry problem-solving benchmarks.

Keywords

Geometry problem solving; multimodal large language models; chain-of-thought; mathematical reasoning; instruction tuning

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