Concise Geometric Description as a Bridge: Unleashing the Potential of LLM for Plane Geometry Problem Solving

TL;DR

This paper introduces a method that converts visual geometric diagrams into textual descriptions using a trained interpreter, enabling large language models to solve plane geometry problems effectively without joint visual reasoning training.

Contribution

The authors propose a geometric description-based approach using CDL and a specialized interpreter, improving PGPS performance with limited data and avoiding joint visual reasoning training.

Findings

The method outperforms existing MLLMs on multiple datasets.

Using CDL matching rewards enhances CDL generation quality.

Training on only 5.5k data achieves competitive results.

Abstract

Plane Geometry Problem Solving (PGPS) is a multimodal reasoning task that aims to solve a plane geometric problem based on a geometric diagram and problem textual descriptions. Although Large Language Models (LLMs) possess strong reasoning skills, their direct application to PGPS is hindered by their inability to process visual diagrams. Existing works typically fine-tune Multimodal LLMs (MLLMs) end-to-end on large-scale PGPS data to enhance visual understanding and reasoning simultaneously. However, such joint optimization may compromise base LLMs' inherent reasoning capability. In this work, we observe that LLM itself is potentially a powerful PGPS solver when appropriately formulating visual information as textual descriptions. We propose to train a MLLM Interpreter to generate geometric descriptions for the visual diagram, and an off-the-shelf LLM is utilized to perform reasoning. Specifically, we choose Conditional Declaration Language (CDL) as the geometric description as its conciseness eases the MLLM Interpreter training. The MLLM Interpreter is fine-tuned via CoT (Chain-of-Thought)-augmented SFT followed by GRPO to generate CDL. Instead of using a conventional solution-based reward that compares the reasoning result with the ground-truth answer, we design CDL matching rewards to facilitate more effective GRPO training, which provides more direct and denser guidance for CDL generation. To support training, we construct a new dataset, Formalgeo7k-Rec-CoT, by manually reviewing Formalgeo7k v2 and incorporating CoT annotations. Extensive experiments on Formalgeo7k-Rec-CoT, Unigeo, and MathVista show our method (finetuned on only 5.5k data) performs favorably against leading open-source and closed-source MLLMs.