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

TL;DR

This work reframes PGPS as a two-stage multimodal-to-text problem by training an MLLM Interpreter to convert geometric diagrams into concise CDL descriptions, then leveraging an off-the-shelf LLM for reasoning. It introduces CDL, three-part structure (ConsCDL, ImgCDL, TextCDL), and a two-stage training pipeline: CoT-augmented SFT followed by GRPO with CDL matching rewards, including S_f, S_C, S_I, and S_T. A new dataset, Formalgeo7k-Rec-CoT, supports CDL training and CoT annotations; experiments on Formalgeo7k-Rec-CoT, Unigeo, and MathVista show strong performance with only 5.5k training samples, outperforming open-source MLLMs and rivaling closed-source models. The approach highlights the potential of reformulating visual geometry into structured text to exploit LLM reasoning, with confirmed benefits in CDL conciseness and targeted reward design for robust CDL generation.

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.

Figures (7)

• Figure 1: (a) MLLMs. Though MLLMs can directly perform PGPS, they still suffer from heavy visual perception errors or logical reasoning errors. (b) LLMs. LLMs are not capable of PGPS without access to geometric diagrams. (c) Ours. We employ a MLLM interpreter to convert geometric diagrams into a concise CDL description upon which an LLM solver performs reasoning.
• Figure 2: Method Overview.(a) Data Construction. We construct Formalgeo7k-Rec-CoT by manual reviewing Formalgeo7k v2 and incorporating the Chain-of-Thought (CoT). We design a two-stage pipeline to train MLLM Interpreter, including a (b) CoT-Augmented SFT Stage and a (c) GRPO Stage with CDL Matching Rewards. Based on the generated CDL, an (d) LLM Solver directly performs reasoning and derives final solutions.
• Figure 3: Qualitative Results on Formalgeo7k-Rec-CoT. We present an example illustrating the complete pipeline of CDL Solver: an MLLM Interpreter converts geometric inputs into CDL upon which the LLM Solver performs reasoning and derives the final answer.
• Figure 4: Illustration of CDL's Conciseness. A textual description of a geometric diagram can be decomposed into shape descriptions ($H$), relation descriptions ($R$), and irrelevant words ($O$). We use "basic" to denote the triangles that cannot be decomposed further, and "composition" to denote the complex triangles composed of two basics. Under specific constraints in the CDL rule, $|H^{\text{C}}|$, $|R^{\text{C}}|$, and $|O^{\text{C}}|$ are minimal to $|H^{\text{T}}|$, $|R^{\text{T}}|$, and $|O^{\text{T}}|$, respectively. Therefore, for any given geometric diagram, a CDL description is significantly more concise than a general textual description.
• Figure 5: Qualitative Result on Formalgeo7k-Rec-CoT.